Note: Links with an asterisk (*) indicate that the entry has not yet been made for that item. This glossary, like the text, is an ongoing project.
Acceleration is the rate of change of velocity. (Here, I am talking about the acceleration of a point - either an object considered as a point or some point on an object. For an extended object in rotational motion, there is angular acceleration to discuss.)
Acceleration is a vector, so it must be given a direction as well as a magnitude. For motion in a straight line, this means assigning acceleration a positive or negative sign. There are four possibilities for acceleration in straight-line motion for determining the sign. The object can be moving in the positive direction and speeding up, the object can be moving in the positive direction and slowing down, the object can be moving in the negative direction and speeding up, or the object can be moving in the negative direction and slowing down. When the object is moving in the positive direction and speeding up or when it is moving in the negative direction and slowing down, the velocity is getting more positive (or less negative - same thing), and the accelaration is positive. Whan the object is moving in the positive direction and slowing down or when it is moving in the negative direction and speeding up, the velocity is growing less positive (or more negative), and the acceleration is negative.
For example, if you are traveling in the positive direction and speed up by 5 m/s in 2 s, the (average) rate of change of your velocity is 5 m/s / 2 s = 2.5 m/s2 (2.5 meters per second per second), which is also your acceleration. If you slow down by 5 m/s in 2 s, the rate of change of your velocity (and your acceleration) is -2.5 m/s2.
The figure shows two of the above cases graphically, by plotting the velocity of a body moving in a straight line versus time. Since acceleration is the time rate of change of velocity, it is also the slope of the body's velocity versus time curve. At time 1 the body is moving in the negative direction (v1 < 0) and speeding up (negative acceleration, indicated by the negative slope of the tangent line). At time 2 the body is still moving in the negative direction (v2 < 0) but slowing down (positive acceleration, indicated by the positive slope of the tangent line).
For the motion of a body along a curved path, acceleration has two components. Tangential acceleration is the time rate of change of the magnitude of the velocity of the body, i.e. the time rate of change of the speed. The other component is centripetal acceleration, which is due to the change in the direction of the velocity. The tangential acceleration is always either in the direction of motion (speeding up and positive) or opposite to it (slowing down and negative). Centripetal acceleration is perpendicular to the direction of motion (and therefore perpendicular to the tangential acceleration) and points toward the inside of the curve the object is moving through. Numerically, it is equal to the square of the speed divided by the radius of curvature at the point along the path where the object is located.
Accuracy is not the same as precision. Accuracy has to do with how close you are to a certain value. For example, you may have measured a property, say your weight on the bathroom scale, several times and got a range of values. The closer your average is to the "correct" value, the more accurate it is. The individual measurements may be all over the place, but your result is accurate so long as the average stays close to the "correct" value as more and more measurements are taken.
General usage: the state of having an ample amount, as in, "Anna Nicole Smith was a woman of great amplitude." Physics usage: the farthest (usually measured in distance or angle) an oscillating body or a system is removed from its equilibrium position during its oscillation.
Whenever authors talk about vague historical figures we use the term "ancients". This way we don't actually have to look up any specific information about who they were and what they actually did. Quite often the term "ancients" appears to mean the classical Greek civilization. These "ancients" were really old but important Greek people like Melina Mercuri.
The earth is extremely old, some 4.55 billion years. So you would think the "ancient world" would refer to at least 3 billion years or so in the past. However, we humans have such a short life span that the ancient world, to us, was the time before fast food. Jimmy Carter is ancient. Linda Byrd Johnson is ancient. Pee-wee Herman is merely medieval. I pretty much take the ancient world to be anything I can't personally remember.
Just like translational acceleration is the time rate of change of translational velocity, angular acceleration is the time rate of change of angular velocity. In freshman/sophomore physics the discussion of angular velocity and acceleration is usually confined to objects rotating in a plane (such as a roulette wheel) and objects moving in a fixed circle or circular arc (such as a pendulum bob). For these objects angular acceleration is quite analogous to linear acceleration.
For motion in a straight line there are four possibilities: The object can be moving in the positive direction and speeding up (positive acceleration), moving in the positive direction and slowing down (negative acceleration), moving in the negative direction and speeding up (negative acceleration), or moving in the negative direction and slowing down (positive acceleration).
Rotational motion in a fixed circle or circular arc is similar to motion in
a straight line as follows. Take a wheel, (but not from my car, please). One
direction of rotation is chosen to be
positive (usually counterclockwise) and the other negative. The wheel can be
rotating in the positive direction and increasing its speed (positive angular
acceleration), rotating in the positive direction and slowing down (negative
angular acceleration), rotating in the negative direction and speeding up
(negative angular acceleration), or rotating in the negative direction and
slowing down (positive angular acceleration). In calculus notation the angular
acceleration (α) is expressed as the time rate of change of the angular
velocity (ω) as follows.
α = dω/dt.
Since angular acceleration has a direction, it is a
vector. However, it is not the same sort of vector as
translational acceleration; it is an axial vector.
You curl the fingers of your right hand in the positive direction of the
angular acceleration, and your thumb will point in the direction of the
angular acceleration vector. The magnitude of the vector will be the magnitude
of the angular acceleration.
This is defined for a body that can rotate about some axis. The angle through which it has rotated is its angular displacement. Angular displacement can be either positive or negative with the direction of positive rotation entirely arbitrary (except that, by convention, a counterclockwise rotation is often considered positive). Typically, angular displacement is measured in revolutions, degrees, or radians. The radian is the SI unit of angular measurement.
Angular displacement is an axial vector. Curl the fingers of your right hand in the direction of the displacement, and your thumb will point in the direction of the angular displacement vector. The magnitude of the vector will just be the amount of the displacement.
Not to be confused with the impetus driving a fisherman toward a body of water, which is called "angler momentum". (This type of momentum is epitomized by the bumper sticker: "My wife told me if I went fishing one more time she'd leave me. God, I'll miss her!") Angular momentum is measure of rotational inertia, just as linear momentum is a measure of translational inertia. Angular momentum must be measured about either a point or an axis. A body rotating about some axis will have an angular momentum with a magnitude equal to its moment of inertia about that axis times its angular velocity.
Angular momentum is an axial vector (in contrast to your usual run-of-the-mill translational vector). The direction of this vector is parallel to the axis of rotation. Curl the fingers of your right hand around this axis in the direction of rotation, and your thumb will point in the direction of the angular momentum vector. The magnitude of this vector is the magnitude of the angular momentum.
The angular momentum of a point mass (or the center of mass of an extended object) that is not necessarily in rotation can nevertheless be defined with respect to a point. Let the direction of the instantaneous velocity of the object define a straight line extending both in the forward and reverse direction. Draw a line from the point about which the angular momentum is to be measured perpendicular to this line. The magnitude of the angular momentum is equal to this distance times the linear momentum of the object.
The direction of the angular momentum follows the right-hand rule. Extend your index finger in the direction of the perpendicular line defined above as drawn. Point your middle finger at someone you dislike - check that - point your middle finger in the direction of the linear momentum. Your thumb will point in the direction of the axial vector of the angular momentum.
There was a parade in our small town some time ago in which little girls who belonged to a local church group paraded with halos and wings. The banner that was carried in front of them read, "GOD'S LITTLE ANGLES". We were chuckling at this apparent misspelling, when, a moment later, this 7° angle ran by, then a 5° angle, then a 9° angle. As even more small angles ran by, it dawned us that the church group had stolen the angles' banner! I mean, what is the world coming to!
Apparently, the angles finally realized what had happened to their
banner and raced through the parade to retrieve it. This is one example
of angular velocity. Another is the rate at which an angle is changing. If an
angle θ changes by an amount Δθ in a time Δt, the
average angular velocity is
ωav = Δθ/Δt.
Instantaneous angular velocity refers to the angular velocity at a specific
time and is the "infinitesimal" change in the angle dθ divided by the
corresponding "infinitesimal" elapsed time dt:
ω = dθ/dt.
Angular velocity is an axial vector
whose direction can be defined using a right-hand rule.
(Curl the fingers of your right hand in the direction of the circular
motion and your thumb will point in the direction of the angular velocity.)
However, in the case of rotation that remains in the same plane, the vector
property is usually just designated by positive or negative values. One
rotational direction is taken as positive (often this is counterclockwise) and
the other is taken negative. The magnitude of the angular velocity is the
angular speed.
When you stand on your bathroom scale and disgustedly look at the readout, you are checking out your apparent weight, which is equal to your actual weight so long as your bathroom is not in, say, an elevator. If your bathroom is in an elevator (think of the plumbing problem that would pose) then what your scale reads is not your actual weight unless the elevator is not moving or is rising or falling at a constant speed. However, you may get a heart-stopping shock if the elevator is accelerating upwards, because the reading on your scale may be much greater than your actual weight. On the other hand you may be in for a pleasant surprise if the elevator is accelerating downward. In this case the scale reading will be less than your actual weight. This effect is the same as the feeling of heaviness or lightness you have in a quickly accelerating elevator. Apparent weight is the reading of a weight scale. This may not be the same as the actual weight of the object, which is the force of gravity on that object. See also weightlessness.
Archimedes was one of the greatest scientists of ancient times as well as a successful and prosperous banker. He came up both with the principle by which the buoyant force could be calculated as well as the idea of lending money and charging interest on the principle. You therefore must be careful to specify which "principle" is meant. The confusion of the two is the origin of the expression "to float a loan". His buoyancy principle can be stated like this: The buoyant force acting on an object emersed either partially or wholly in a fluid equals the weight of the displaced fluid.
What could keep you out of the NFL draft if you are a wide receiver. However, if you are a physics student this means the average rate at which an object covers distance. This could be defined over a time interval or between two points along the object's path. The formula for average speed is distance covered divided by the time taken to cover that distance. Speed is always positive and is a scalar quantity.
An axial vector is different from a "regular" vector in one crucial way. Both types of vectors have magnitude and direction. Both can be represented by arrows. However, an axial vector has to do with rotation whereas a "regular" vector has to do with translation (changing position). This, weirdly enough, means an axial vector is not reversed by a mirror like a "regular" vector is.
Axial vectors include angular displacement, angular velocity, angular acceleration, and angular momentum. To find the direction of an axial vector you employ a right-hand rule. You curl the fingers of your right hand in the positive direction defined for the angular quantity. The thumb of your right hand then points in the direction of the axial vector.
The feeling a physics student experiences when she finds out the test has been postponed a week. (This is followed by some serious partying and a sinking feeling when the test week inevitably comes around. Not to mention the 10 points off her paper for missing the problem involving buoyant forces.) For the information of our slacking student, buoyancy refers to the force acting on an object in a fluid (or even in a solid that acts like a fluid over a long period of time) arising from hydrostatic pressure. Hydrostatic pressure, in turn, is caused by the force of gravity. Buoyancy is responsible for the upward force on an object placed in a fluid (liquid or gas), causing wood to float in water and a metal object, say an anchor, to sink less rapidly than it would in air. The amount of the force is given by Archimedes' principle.
There is no truth to the rumor that the British system of units was invented by the Three Stooges. Many scholars have cited the legend that it was imposed upon the Saxons by the conquering Normans as a form of humiliating punishment.
However, recent research has shown these units were actually invented by Monty Python as a satirical gag. Unfortunately, the House of Lords got a copy of the script, and, not getting the joke, thought it made jolly good sense to adopt it for the entire British Empire. The fundamental mechanics units are therefore as follows. Length is defined by John Cleese's foot, hence the name. (Cleese is a great fan of Tex-Mex food. This is how the "inch", short for "enchilada", came to be.) The time unit is called the "second", because the troupe decided to reject their first effort at defining this unit. Finally, they named the unit of mass the "slug" after those giant weights that fell down on Monty Python characters.
The amount of shrinkage the Bulk experiences depends on the
amount of hydrostatic pressure applied. This is true
of most ordinary substances. If a body or substance is
elastic, the magnitude of the change in volume divided by the original
volume (that is, the fractional change in volume)
is proportional to the pressure. However, the change in volume is negative (the
volume gets smaller), so you must provide a negative sign when you turn this
statement into an equation by supplying a constant of proportionality. The
constant of proportionality is called the bulk modulus.
The equation defining the bulk modulus, B, is
ΔP = -B(ΔV / V),
where ΔP is the change in pressure, ΔV the change in volume, and V
the volume at the original pressure. B has the units of pressure. The larger
the bulk modulus, the more difficult it is to compress a body or substance.
Gravity is a body force, which means it acts on every atom in whatever body is experiencing the force. (That would be every body in the universe, as far as I know.) It would therefore appear to be ridiculous to think there might be one point in the body where the force of gravity can be considered to be acting. And it is, because then the body would be grotesquely distorted by having all its weight concentrated at this one point.
However, the idea does have merit, it turns out, if you are only concerned about the motion of this point, called the center of gravity. Mathematically, so long as you are not concerned about the rotation or deformation of an object, you can consider that all its mass is located at this one point and treat the object as a point that experiences the entire gravitational force. For example, to compute the force of gravity between the earth and the moon, you need to know their masses and how far apart they are. Mathematically, this distance turns out to be the distance between their centers of gravity, which, to a good approximation can be taken to be their geometric centers. Then you compute the mutual orbit of the earth and moon due to their mutual gravitational attraction by computing the motion of their centers of gravity.
Your center of gravity can't be deduced so easily. What we can do is stick you on a sharp pole and adjust your position until you are balanced, taking care not to puncture you, since, like a wiener on a stick, you could then be held on the pole without having to be in balance. The balance point is your center of gravity. The concept of the center of gravity is closely related to that of the center of mass.
The center of mass is easily defined for a system of,
say, N particles. Imagine these particles are all along a straight line, the x
axis. For this linear array of (point) particles the center of mass along the x
axis is defined mathematically as
xcm = ∑ mi xi / ∑ mi.
In this formula "i" (the "index") numbers the masses, "mi" is the
mass of the i'th particle, and xi is its
position along the x axis. The sum is taken over all N
particles. You can let the index be implied and write this equation a bit more
economically as
xcm = ∑ mx / &sum m.
If the particles are spread out over three-dimensional space, you can come up
with equations for ycm and zcm in the same way. The
center of mass for the collection of particles is
(xcm, ycm, zcm). These equations
also work for extended bodies if you do the sum over their atoms. (For bodies
of high symmetry, this means performing an integration.)
The center of mass of an object is the same as its center of gravity if the object is in a uniform gravitational field, which is nearly always the case (at least as a close approximation). To picture a uniform gravitional field, think of the situation where the weight of an object is the same in magnitude and direction at any position in a region of space. This region of space then contains a uniform gravitational field. (The gravitational field over a relatively small area of the earth's surface is pretty much uniform. You are not going to weigh more or less at the back of the physics lecture room than at the front. Neither is the pull of gravity going to be in a significantly different direction.) In such a situation you can find the center of mass by the same balancing act used for the center of gravity.
"Centripetal" means "toward the center", as in, "Politicians of the left and right tend to conduct centripetal campaigns when running for broader office." Centripetal acceleration is due to the change in the direction of an object's velocity, not its speed. Strangely enough, this acceleration is perpendicular to the actual direction of motion, very much like your left- or right-wing candidate, whose "center seeking" does not always reflect his or her actual tendencies.
Centripetal acceleration is experienced by any object that travels in a curved path. It is directed toward the concave side of the curve the object follows, perpendicular to its velocity. For circular motion, therefore, centripetal acceleration is always pointing toward the center of the circle. In spite of this, however, the object in circular motion never gets any closer to the center. Again, this is remarkably like the campaign of politicians for whom getting elected is more important than being honest with the voters. After they are elected you find that, rather than being toward the center of policy as you believed, they remain solidly in the camp of their most generous campaign contributors.
Finally, centripetal acceleration has a
value equal to the square of the speed of the object, v, divided by its
instantaneous radius of curvature, r (or the radius of the circle if the
motion is circular). Symbolically,
ac = v2 / r.
The "instantaneous radius of
curvature" is the radius of the circle the object would travel in if it kept
turning at the same rate. Here is where the resemblence to politicians ends.
The force that causes centripetal acceleration. Since this force is responsible for an acceleration, it must be a net force. As such, it might be a single force, such as the the tension on a rock tied to a string and twirled in a horizontal circle. If the rock is twirled in a vertical circle, the compenent of the force of gravity acting in the same direction as the string also contributes to the centripetal force, illustrating that the centripetal force may be a combination of two or more forces.
It would be easy to say this was the organization that Maxwell Smart did battle with and leave it at that. It would be so much simpler. Chaos, in physics, however, is not at all simple. The behavior of chaotic systems cannot be predicted indefinitely. No matter how accurately you can characterize a system at some point in time and no matter how well you understand the physics of such a system, eventually the behavior of the system will depart more and more from your predictions. Large changes in the behavior of a chaotic system can arise from resonances. Examples are the top, the obliquity of Mars, and the atmosphere. Mathematically, chaotic behavior arises from highly complex ("nonlinear") equations that govern the physics of such systems. Solve the future of a chaotic system using these equations with two sets of initial conditions with the tiniest of differences, and you will come up with two entirely different futures.
Studies involving force and motion in physics are termed "mechanics". This has led to a lot of silly jokes about, for example, a quantum mechanic repairing an atom, or a classical mechanic fixing Beethoven's piano. I don't believe any mechanics were called "classical" until relativistic mechanics and, especially, quantum mechanics came along. It may sound as if classical mechanics is outdated - essentially the study of matter based on Newton's laws of motion, which are over 300 years old. However, it is still extremely important, being the major vehicle used by college professors to chase off freshman physics and engineering students. And it does have some nonacademic uses, for example in cars, trucks, ships, planes, construction equipment, factory machinery, pumps, cooling and heating, oilfield equipment, water treatment, weapons design, amusement parks, etc. There are even a few academics still working on basic problems in classical mechanics. Newton lives!!
A collision is something running into something else, which happens very often in nature due to the absence of air traffic controllers. (God fired them all when they tried to unionize.) Left to chance, collisions are much more likely between two objects than between three or more, so I will concentrate on two-body collisions.
Energy-wise, a whole range of possibilities exists for colliding objects. They might bounce of each other without, on the whole, losing kinetic energy. That is, whatever kinetic energy is lost by one object is gained by the other such that the total kinetic energy is the same. This is called an elastic collision, not necessarily because the two bodies are elastic but because elastic bodies do not lose kinetic energy when they collide and are therefore used to typify this type of collision.
The type of collision in tune with family values is the completely inelastic collision. In this collision the bodies exchange vows, become "one flesh" (merge), and move together after the collision. This ceremony results in the greatest loss of total kinetic energy possible, given the initial momenta of the colliding bodies. (This is largely due to the domestication of the male member of the union.) The lost kinetic energy is eventually dissipated as heat (at least during the honeymoon).
Most collisions are neither elastic nor completely inelastic. In these licentious encounters there is an exchange of kinetic energy, some of which is lost but not as much as if the collision had been completely inelastic. In physics the term "collision" usually refers to an encounter that is short enough and intense enough that the forces acting on the colliding bodies other than the forces of the collision itself are relatively small and therefore can be ignored.
See the definition of linear momentum. The rate of change of linear momentum of an object with time is equal to the sum of the forces acting on that object. (This is Newton's second law of motion). If the forces sum to zero, then the linear momentum does not change: the object continues in the same direction with the same speed (assuming it is neither losing nor gaining mass - this is Newtons first law of motion).
If you are talking about a system of several objects, the sum of the linear momenta of all the objects remains the same if the sum of all the external forces acting on the objects is zero. Two points of possible confusion need to be addressed: (1) The momenta of the individual objects do not necessarily remain the same, and, in fact, usually change. It is the sum of all the momenta that stays the same. (2) Forces acting between objects in the system don't change the sum of the momenta. These forces are equal and opposite inside the system and cancel out. These forces may, however, change the momenta of the individual objects (but never the sum). Conservation of linear momentum thus states that if the external forces acting on a system sum to zero, the total momentum (sum over all the objects) remains constant. This rule is useful, for example, in analyzing collisions.
These forces belong to all-male clubs, where they sit around smoking pipes and saying, "Hrrummph!", a lot, especially when reading the newspaper. They play chess and checkers, watch Lawrence Welk reruns, and subscribe to "The Wall Street Journal". However, we are not interested in their social lives. These forces don't do something that makes them remarkable: They don't do any work on an object that is moved from some position, through any path whatsoever, possibly around all creation, then back to the original position. Such a path is said to be "closed", and when you sum up the work done on an object by a conservative force over a closed path, it is zero.
A conservative force depends only on the positions of the objects which interact with each other by means of this force, which means a potential energy can be defined for it. Because the work such a force does around a closed path is zero, it follows that the work it does on an object moving from some initial point A to some final point B is independent of the path taken between A and B. When you bring the work-energy theorem into this, you find that the change in kinetic energy of an object moving between point A and point B with only (one or more) conservative forces performing work on it is independent of the path taken. Finally, such a force is called "conservative", because the previous statement implies that if an object moves around a closed path with only conservative forces doing work on it, its kinetic energy will be the same when it returns as it was when it began, that is, the kinetic energy will be conserved for the round trip.
A conversion factor is a ratio of units of measurement that allow you to convert from one unit of measurement to another, such as from fathoms to meters. The units in the ratio must be the same type of measurement, such as time over time or area over area. Conversion factors are always equal to the number one with no units! They have to be for the measurement to be the same amount. For example, two feet is about 0.71 m, the same distance in either unit, found by multiplying 2.0 ft by the conversion factor 0.3048 m/ft.
A coordinate system is "simply" a way to specify location. Coordinate systems are important in surveying, mathematics, navigation, wars, football, and physics. A coordinate system needs an origin, which is the arbitrary location from which to measure distance. In a cartesian coordinate system straight lines are drawn from the origin to define direction. These lines are called axes, and, on a two-dimensional sheet of paper, there are two of them drawn at right angles to each other to make a cartesian coordinate system.
One axis, usually called the x axis, is often used to measure the right-left position of a point. The other axis, often called the y axis, would then measure position up and down. A position to the right of the origin along the x axis is usually considered positive and to the left is negative. Above the origin on the y axis is positive and below is negative. A position located at a point that is not on an axis is a combination of a positive or negative x and a positive or negative y. Therefore the position of a point can be expressed as, for example, (3.5, -1.2), which means 3.5 distance units (e.g., cm) to the right of the origin and 1.2 units below the origin.
If you need to measure position in three dimensions using a cartesian coordinate system, you have to "draw" a third axis, the z axis, from the origin perpendicular to both the x and y axes. These axes can be viewed as forming one corner of an imaginary box. There are other coordinates systems useful in particular circumstances, such as polar coordinates, cylindrical coordinates, and spherical coordinates.
This is not a real force, but what is referred to in physics as a "fictitous" force or "inertial force. Try to find one of those old-fashioned turntables used to play vinyl records. Take a marble, coat it with paint, and place it on the turntable. Assume there is no friction between the marble and the turntable. This means that as the table turns the marble stays put due to its inertia and creates a circle of paint at its radial position. Now give the marble a push directly toward the edge of the turntable. In your frame of reference, the marble will move in a straight line due to the absence of friction to change its direction of motion. However, it will trace out a curved path of paint on the turntable. From the perspective of the turntable, the marble has not moved in a straight line, but in a curved one. Assuming it takes a force to change the direction of motion of the marble, an observer on the turntable would conclude that some force - the coriolis force - has acted on the marble.
The problem for the observer on the turntable is that she is in an accelerating frame of reference (accelerating by centripetal acceleration). In a non-accelerating reference frame, the marble moves in a straight line, as no actual force is acting on it. In the accelerating frame, however, the marble appears to take a curved path and therefore be acted upon by a force. Accelerating reference frames give rise to fictitious forces - forces that seem to be acting to those in the accelerating frame, but not to those in non-accelerating frames. The coriolis force is the fictitious inertial force that seems to make things move along curved paths as seen by someone in a rotating reference frame. The coriolis force is the reason the air in atmospheric pressure systems flows either clockwise (high pressure systems in the northern hemisphere and low pressure systems in the southern hemisphere) or counterclockwise (low pressure systems in the northern hemisphere and high pressure systems in the southern hemisphere). Air moving on a rotating earth will appear to move in a curved path.
Why did the chicken cross the road? Well, perhaps the chicken was
treacherous. What if you were to cross the chicken with the road? Around
here you would probably get a chicken full of pot holes.
On the other hand, if the chicken and the road are both vectors, you would get
a third vector pointing at right angles to both the road and the chicken.
That's because the cross (or
vector) product is a way of multiplying two vectors to get a third vector. If
A and B are vectors with magnitudes A and B, then the magnitude
of the vector product, C, is
C = ABsinθ,
where θ is the angle between the two vectors. The direction of C is
given by the right-hand rule. Point your index finger in the direction of
A, your middle finger in the direction of B (hoping, of course,
this doesn't tick B off), then your thumb points in the direction of
C, perpendicular to both A and B. (See
Figure 8.14.) The vector equation for
this operation is
C = A × B (= -B × A).
Note that the cross product anticommutes.
This could be a hot topic if it were about Paula Abdul. Unfortunately for you male students, we don't have time for such tawdry excursions. Curvilinear motion is like the TV car commercials where the featured product is zipping around curves with leaves flying behind it.
For curvilinear motion you need
either a point object or a point on an object (such as the
center of
gravity) moving along a curved path (although, motion in a straight line
can be considered a special case of curvilinear motion). The velocity of such
a point is always tangent to its path and in the direction of motion. (Well, of
course! The velocity is what determines the direction of motion!) You
can divide up the acceleration of the point into a
tangential and a centripetal component. The tangential component is tangent
to the path and either in the direction of motion (speeding up) or opposite
the direction of motion (slowing down). The tangential
acceleration is given by the rate of change of the speed. In calculus notation,
at = dv/dt ,
where v is the speed of the point in curvilinear motion. If the object is
speeding up, the tangential acceleration will be positive; if slowing down,
negative.
The centripetal acceleration is that
of an imaginary point that is moving at the instant chosen exactly like the real
point, except that the imaginary point continues curving at the same rate and
so moves in a circle. The magnitude of the
centripetal acceleration is
ac = v2 / r,
where v is defined as before and r is the radius of the circle of the
imaginary point. The definition of the motion of the imaginary point gives it,
instantaneously, the same acceleration, tangential
and centripetal, as the real point. After that instant the accelerations of
the two points are no longer identical. You have to define a separate imaginary
point for each position along the curved path of motion in order to evaluate
the centripetal acceleration of the real point along the path.
This is the extension of polar coordinates to three dimensions. All you do is define an axis (the z axis) perpendicular to the polar coordinate plane. A point in three-dimensional space is then given by the coordinates (r, θ, z). See the figure.
This is a calculus term. The word "derive" means to get something from something else, such as, "Math teachers derive much pleasure from watching their students squirm during a calculus test." A derivative, graphically, is just the slope of a curve at a particular point, see the figure. It is useful in physics because it gives the rate at which one quantity, the one plotted on the vertical axis, changes with a change in a second quantity, the one plotted on the horizontal axis. Lots of quantities in physics are rates, and especially useful are time rates like velocity (time rate of change of position), acceleration (time rate of change of velocity), and physics anxiety (time rate of change of grade point average).
Most quantities are measured in combinations of fundamental units, such as "meters per second", which is the division of one fundamental unit (the meter) by another (the second). Sometimes, for convenience, scientists decide to christen a combination of fundamental units with a name and it becomes a "derived unit". The newton (kg-m/s2), a measure of force, is a common derived unit of mechanics. So is the unit of energy, the joule (kg-m2/s2).
Displacement is the term used for a change in position. For example, "Senator Fuzzworthy's position on the environment underwent a displacement to the right after meeting with the energy lobbyist, who reminded him of the cost of his upcoming re-election." Another example: "Step forward, Tin Man!!". A final example: You are at x = 5 m and y = -2 m in a cartesian coordinate system. When you move to a new position at x = -3 m, y = -4 m, you have undergone a displacement Δ x = -3 m - (+5 m) = -8 m and Δ y = -4 m - (-2 m) = -2 m, that is, 8 m in the negative x direction and 2 m in the negative y direction. The "Δ" can be read "change of" and a change is always "final value minus initial value".
Also known as the scalar product. This is a way to multiply two vectors
together and get a scalar. For a vector A and a vector B,
differing in direction by the angle θ, the dot product is defined as
A•B = ABcosθ,
where A and B are the magnitudes of the respective vectors.
This type of potential energy is due to the elastic force, which is a position-dependent (and hence conservative) force, depending only on the deformation of the elastic body. All schoolboys know that a rubber band can store enough energy to propel a spitwad into the teacher's rear end. This is a case of elastic potential energy being converted to kinetic energy through the work done by the rubber band on the spitwad. Elastic potential energy is proportional to the square of the deformation. In the case of a rubber band stretched an amount x, the energy would be (1/2)kx2, where "k" is called the elastic constant.
The meaning for this term in physics is close to how you view the band in your underwear that prevents you from the embarrassment of having it slip down and show beneath your shorts (or, for some these days, the embarrassment of having it slip down into your shorts). The elastic band in underwear snaps back to its original length when stretched (unless you are talking about my underwear, which has not seen its original shape in years). An elastic substance will regain its original size and shape after deformation as long as its elastic limit is not exceeded.
If a substance (or system) resists being stretched, compressed, or otherwise distorted in shape, the force that acts to bring it back toward equilibrium is called the restoring force (equal and opposite to the force applied due to Newton's third law of motion). If this force only depends on the change of position due to the deformation, it is a conservative force. If the magnitude of the force is proportional to the magnitude of the displacement, it is an elastic restoring force and is proportional to the deformation through an elastic constant. (For a spring this would be F = -kx, where x is the amount of stretch or compression and k is the elastic, or "spring", constant.)
All conservative forces give rise to potential energy, in this case elastic potential energy. This means that perfectly elastic behavior results in no loss of mechanical energy.
An ellipse is an oval-shaped curve, but not just any old oval-shaped curve.
The outline of an American football, for example, is not an ellipse. (Unless
you look at it nose on, in which case the outline is a circle, which is
a special type of ellipse.) To make an ellipse, stick two pins in corkboard
and tie the ends of a string around each pin, where the string will be longer
than the separation of the pins. Take a marker and use it to stretch the
string so it is taut. Now, keeping the string taut, draw a closed curve. You
will have created a curve that mathematically can be expressed by the following
equation.
x2 / a2 + y2 / b2 = 1.
This is an ellipse centered on the origin and oriented so that the axes (which
divide the ellipse into halves) are parallel to the x and y directions. The
axis in the x direction has a diameter of 2a and the axis in the y direction
has a diameter of 2b. If a = b, you have a circle. The longer axis is called
the major axis and the shorter is the minor axis. For example, if a > b,
the major axis will be in the x direction. Another property of the ellipse is
its eccentricity. An ellipse that spends its days in the park feeding
candy-coated nuts to squirrels is possibly eccentric, but this term
actually applies to how round or "flat" the ellipse is. The greater the
eccentricity, the flatter the ellipse. A completely flat ellipse has an
eccentricity of one. A circle is an ellipse with an eccentricity of
zero.
The old definition of energy as "the ability to do work", which perhaps hails from the study of heat engines in the nineteenth century, doesn't really cut it. It does not explain what energy really is, which is OK, because we don't know what it really is. Hence, with energy we have another concept, along with mass, space, time, etc., that we can quantify but not explain. With so many things physicists deal with not being fundamentally understood, you might understandably wonder what good are physics and physicists. The answer is simple as far as I'm concerned. Physics is good because I, as a physicist, can make a living at it, and physicists are good because they can make a living doing physics.
Back to what energy is, things were made even more confusing when Einstein found that energy and matter can be interchanged, thus blurring the distinction between them. As a beginning physics student, it is probably a good idea to content yourself with learning what each species of energy is like in order to get a feel for what this concept means. (See kinetic energy and potential energy, for example.) Then, if you can put it all together in a coherent and fundamental way, please let me know. Time marches on and I have yet to win the Nobel prize.
Nature, like a lazy couch potato, wants to conserve as much energy as possible. Actually, nature wants to conserve all its energy and does a good job of it. Hence, nature is the ultimate couch potato. This is comforting to know while kicking back and watching baseball on TV. My wife does not understand that I am not just watching a sport, I am in a deep communion with the spiritual meaning of the cosmos. (This is no excuse for not doing your physics homework, however.)
Energy comes in many different forms which can be freely interchanged with no loss. (However, there is a loss in useful energy, as described by the second law of thermodynamics.)
The law of conservation of energy also comes in many forms. One of the most useful is when a system's energy consists of only kinetic energy plus potential energy, which occurs when only conservative forces are performing work on the system. In this case any loss of kinetic energy is gained as potential energy and vice versa, making the solution of some otherwise difficult problems fairly easy. In the real world, some energy is almost always "lost" as heat energy. (It's not really lost, it's still there but not all of it can be recovered to do work. See, I told you nature was lazy.)
When the special theory of relativity was developed, it became clear that energy and mass could be interchanged. (This is not why you keep getting fatter while consuming "energy" bars; however, it is the basis of nuclear physics.)
Socially, this is extremely difficult to achieve, but in physics it happens all the time. During the cold war an equilibrium of sorts existed between the United States and the Soviet Union. They could wipe us out and we could wipe them out. This was called "mutually assured destruction" or, more feelingly, "the balance of terror". Equilibrium is a balance, and in mechanics there are two types of balance that can be achieved. If forces are in balance in a particular direction, translational equilibrium is said to exist in that direction.
You can have translational equilibrium in one direction but not another. A child accelerating down a straight, slippery slide is not in equilibrium as far as the motion parallel to the slide is concerned, but is in equilibrium perpendicular to the surface of the slide. In fact, the child is also in translational equilibrium sideways to the slide. Or, consider a drag racer accelerating down a track. If the track is taken as the x direction, the dragster will not be in equilibrium in that direction. However, if it is moving in a straight line, equilibrium exists in the y direction (side to side) and in the z direction (up and down)
The second type of equilibrium is rotational equilibrium. In this case the torques are in balance and the object is either not rotating or is rotating at a constant rate (angular velocity). That is, the angular acceleration is zero. Since a body can rotate about at most three independent axes, it is possible for a body to be in rotational equilibrium about one axis but not another. Consider a child pushing a merry-go-round. If the merry-go-round is speeding up, it is not in rotational equilibrium about the vertical axis through its center. However, it is not rotating at all about any horizontal axis, so rotational equilibrium exists about any horizontal axis you care to consider.
Finally, you can have stable or unstable equilibrium. If the object tends to return to its original state when disturbed, that is stable equilibrium. Consider a marble at the bottom of a round bowl, for example. On the other hand, a coin balanced on its edge is in a condition of unstable equilibrium. You can have both at once, in what is known as the saddle situation. Short cowboys have this problem. Since their feet can't reach the stirrups, it is easy for them to slide off the side of the horse. However, the ridges on the saddle keep them from sliding off the front or back. Since short cowboys were not much good at cutting or bronc riding in the old west, they were primarily used as decoys for Indians or rustlers, which led to a severe short-cowboy shortage and the end of the cattle drives.
Extensive quantities depend on the size of whatever it is you are talking about. Volume, mass, weight, energy, heat, ... these are all extensive quantities because they depend on the scale. On the other hand, intensive quantities do not depend on scale. Mass density, for example, depends on the substance involved, the temperature of the substance, and the pressure it is subjected to, but not on the size of the body the substance constitutes. Three cubic feet of lead has more volume, mass, and weight than one cubic foot, yet both objects are made of lead and, under the same conditions, have the same density. Other intensive quantities are, for example, pressure, temperature, and charge* density.
"Newton's first law of motion" was actually Galileo's idea. It could be called the guy law, because it reveals that nature is essentially masculine. I mean, if this doesn't describe a guy, then what does? "An object at rest will remain at rest, and an object in motion will continue at the same speed in a straight line, unless either are disturbed." This also shows us that disturbances in nature are feminine. When guys refuse to stop and ask for directions, it is merely a consequence of the first law of motion. It takes a disturbance in the passenger's seat to counter this natural tendency. The first law of motion is a statement on the property of matter called inertia, another feature of the masculine sex. It is hard to imagine what it would be like if nature were feminine. Probably every particle in the universe would go to the bathroom at the same time, causing the entire universe to collapse into an all-encompassing black hole and bringing an end to existence. So let's hear it for the guys!
Experience has told us to regard a force as a push or pull, but we are subject to numerous other types of forces, many we are not normally aware of. We usually don't think of the floor pushing up on us with a force, and we pretty much forget about the force of gravity until we find ourselves on a tall ladder. In spite of all the types of forces, there are only four fundamental forces known to physics: gravity, the electromagnetic force*, the strong force*, and the weak force*. In physics a force is defined as the influence which, if acting alone on an object with mass, will cause it to accelerate. This is a definition and gives rise to the equation F = m×a, where m is the mass of the object and a its acceleration due to the force F. A more fundamental view of force is that it is the time rate of change of momentum, a view that is the basis of Isaac Newton's second law of motion. Since force has direction as well as magnitude, it is a vector. May it be with you.
When the subject of the free-body diagram is first discussed, pre-med students think it is an anatomical chart you don't have to pay for, and they get momentarily excited. Quickly I crush their expectations by explaining what such a diagram is as far as physics is concerned. To make a free-body diagram, you isolate the body in question from all other objects. You then indicate the forces that are acting on the object by arrows pointing in the direction of these forces.
The free-body diagram is the most important tool that physics students absolutely refuse to employ to solve physics problems. Physics students would rather be weighted down and tossed into a vat of boiling acid than to draw a free-body diagram. Drawing such diagrams must obviously cause the most severe kind of agony and suffering imaginable, otherwise students desperate to pass physics would use them every chance they got. I implore the medical community to invest resources sufficient to track down the cause and produce a cure for this horrible syndrome. With your help, and that of concerned people everywhere, we will triumph and vanquish this debilitating affliction that prevents students from earning decent grades in physics.
The state of an object that is moving solely under the influence of gravity (or under the influence of physics test scores in the case of GPA). Try to get over the word "fall". In physics we include objects tossed upwards as well as dropped objects. We include projectiles. Even satellites and astronauts in orbit. The word "free" implies lack of support or propulsion.
A body or system that exhibits periodic motion moves over the same path again and again. The time it takes for the body/system to make one complete cycle is called the period. The frequency of this motion is the number of cycles completed per unit time. Often this time unit is the second. One cycle per second (as you might observe passing while standing on the sidelines at the Tour de France) is called a hertz (symbol Hz). Periodic waves can also be characterized by frequency. Imagine a ripple in a pond and a cork moving up and down as the ripple goes past. The time it takes for the ripple to make a complete motion is the period of the wave. The number of complete motions made per second would be the frequency in hertz.
Units of measurement defined in terms of standards that can be reproduced by experiments carried out in a laboratory. Fundamental units are those upon which derived units are based.
Think of Mrs. Puff on Spongebob Squarepants.
This is a very weighty matter - a subject you can really sink into. I mean, like, heavy, man. There are tons of things you can say about gravity, and physics instructors, like me, are determined to pound it into our students. Along with a lot of bad puns. Isaac Newton was the first to recognize that all matter attracts all other matter. This attractive force is called gravity, although most students in freshman physics don't think studying it is so attractive for some reason. The attraction grows in strength with the amount of matter (the more matter, the greater the attraction - which doesn't always work that way between human beings, depending on how the matter is distributed on the body). The attraction weakens with distance (the farther away the matter is the less the attraction - again sort of contrary to the case between people since "absence makes the heart grow fonder", although a variation of this saying, "absence makes the heart go wander", is more in tune with physics).
The formula given
by Newton for the attraction between two bodies of masses
m1 and m2 a distance r apart is
F = G m1 m2/r2,
where G is the universal gravitational constant (essentially a constant of
proportionality that depends on the system of physical units), which has the
accepted SI value of
6.673×10-11 N·m2 / kg2.
The British unit for measuring power, the rate of expenditure of energy. The standard for this unit is an aging horse kept at the Royal Greenwich Observatory. The horsepower is defined as the power this horse expends in lifting the royal consort (currently Prince Phillip) up into the air by means of a pulley system attached to the horse's harness. Because of the advanced age of the horse, the horsepower has lost its value against the watt as the horse has grown weaker and Prince Phillip more hefty.
Of course, the British don't use the British system anymore. Instead they maintain the horsepower on contract to American automobile manufacturers, who are afraid that, if their engines are rated in watts instead of horsepower, the buying public will think either that their cars should light up or are powered by steam. The automobile industry would then be inundated with lawsuits by people claiming to have been scalded when the steam spilled into their laps.
In fact, the horsepower is the reason the British finally abandoned their own system of units. Many years ago the standard horse died suddenly and unexpectedly, and immediately, all around the country, engines running on horsepower came to an abrupt halt. The British economy was nearly ruined while metric Europe prospered. This was felt in the U. S. as the so-called "energy shortage" of the 1970's. The American people were told that this shortage was due lowered OPEC production, but we physicists knew that the government was hiding the true cause of the disaster, the death of the standard horse, for fear of creating a panic.
The British swallowed their national pride and went metric, but the
the U. S. government still promulgates the lie of the energy shortage.
Due to the weakening condition of the current standard horse, and the danger
it might die at any moment, there is a movement afoot to train a younger horse
as the new horsepower
standard with the aim of bringing the rate of exchange between the watt
and the horsepower back to its official value, namely,
1 hp = 550 ft·lb/s = 746 W.
This is an open curve given by the equation
x2 / a2 - y2 / b2 = 1,
for a hyperbola centered on the origin with its axis in the x direction.
Ideally, the gravitational path of a small object about a much larger object
follows such a curve. The parameters a and b are constants that are found
when solving for the path of a particular object.
(This would be a great name for a credit card.) Many of you probably think the scientific definition of "impulse" has something to do with propelling starships at subwarp speeds. Not so, trekkies. An impulse is the change in momentum of an object involved in a collision, where a collision is viewed as a short (compared to the overall travel time of the objects involved), intense (compared to the other forces acting on the objects) interaction between two or more (usually two) objects. From Newton's second law of motion, the impulse acting on a body is also the average force of the collision acting on the body times the time over which that force of collision acts.
Apparently, the word "inertia" is related to the word "inert". Both are technical terms used by professors to describe their students and how hard it is to motivate them to learn. Inertia in students is a great mystery, given all the scintillating lectures they are priviledged to hear from some of the most dynamic motivational speakers to ever appear on the face of the earth: your average college professors. So we know it's not our fault. If you think I lie, try to follow the remaining description of inertia without falling asleep.
Inertia really is a great mystery. Why do objects resist a change in their state of motion? If they are at rest, they don't want to move; if they are moving, they don't want to change speed or direction. You can devise experiments to measure the difficulty in moving stationary objects, and these lead to the concept of inertial mass. (See also the discussion in the text.)
Inertial mass is the measure of inertia. That is, the inertia of a 10-kg body at rest is 10 kg. (More generally, we consider relativistic momentum to be the measure of inertia of a moving body, a concept that is intimately related to rest mass in relativity theory.) In sum nobody, not even I, knows what inertia is, although we can, as usual (ho-hum), measure it.
An inertial reference frame is one in which Newton's first law of motion holds. That is, an object set in motion continues moving in a straight line at a constant speed. Or, if it is not set in motion, it just sits there. College students, therefore, make excellent inertial reference frames. The surface of the earth, however, is not an inertial reference frame, because if you throw a baseball, it will turn downward and strike the ground. (At least if you are Chuck Knoblock of the 2000 Yankees trying to throw to first.) However, we fix that by saying it really is an inertial reference frame; it's just that there is this force, gravity, acting in what would otherwise be such a frame. A genuine inertial reference frame is one where gravity is not acting and the observer is not accelerating. These are used extensively in special relativity to analyze the effects of motion on time and space.
A body in motion can be considered to be rotating about a particular point at a particular time. This point must be instantaneously stationary. Perhaps the easiest example to understand is that of a wheel rolling without slipping. The point on the wheel touching the surface on which it is rolling must be instantaneously at rest if the surface is at rest. The wheel can therefore be thought of as rotating about this point at the instant the point touches the surface. At a later instant this point has been replaced by another, different, point on the wheel. Such a point can always be found for a rotating object, although the way of finding it belongs to courses on dynamics more advanced than freshman or sophomore physics. Even a body moving in a straight line can be thought of as rotating about a point an infinite distance away. Now only someone with a truly twisted mind would think this way, and, thankfully, this is a course in physics, not mathematics.
Imagine the following scenario. You are on a race track in a formula one race car. Your job is to illustrate instantaneous speed for all the eager physics students in the stands. You drive down the track, first acclerating, then braking, in order that your speed varies from one moment to the next. You have a buzzer in the car. When a designated student presses a button, the buzzer goes off and you must, from that time for several seconds, maintain a constant speed. This speed is then measured by dividing how far you traveled at the constant speed after the buzzer sounded by the corresponding time elapsed. The result is your instantaneous speed at the sound of the buzzer, because, if you are a skillfull enough driver, that was the speed at the exact moment the buzzer sounded.
You can also view instantaneous speed graphically. If you plot the position of an object (on the vertical axis) versus the time it was at that position (on the horizontal axis) then the slope of this curve at a certain time is also the magnitude of the instantaneous speed of the object at that time. (If you let the slope take its sign, what you have is the instantaneous velocity of the object.) Now, for you techy physics types this is the magnitude of the derivative of the position with respect to time at that particular instant.
Technical physics students should already know what an integral is, so even if they read this, they wouldn't admit to it. For the rest of you poor, uninitiated, non-technical physics folk, I will be simple and precise: Look it up in a mathematical dictionary and quit bugging me! On second thought, perhaps I shouldn't be so lazy and give this a shot. Imagine a curve, any kind of curve. Not that kind of curve! Get your mind back on physics! Draw two vertical lines from the horizontal axis up to (or down to) the curve. Shade the area bounded by the curve, the vertical lines, and the axis. The area of the shaded region is the integral of the curve between the two vertical lines. See the figure. If the shaded region is below the axis, it is negative. If the shaded region contains regions both above and below the axis, then you subtract the area of the region below from that above.
When waves run into each other they will "interfere". You probably have relatives like that, possibly a mother-in-law. The interference of waves occurs because when wave crests encounter wave crests, they build up larger crests; when wave troughs encounter wave troughs, they build deeper troughs; when a wave crest encounters a wave trough, they tend to cancel each other out. When colliding waves enhance their amplitudes, it is called constructive interference. This is not the kind you usually get from relatives. No, relatives usually engage in something like destructive interference, where wave crests and troughs tend to cancel out and destroy the wave action. On second thought, this doesn't make sense, because I've known relatives to make waves by their destructive interference. Sorry, I'm getting confused. Maybe you should consult a different reference for this topic.
The SI unit of energy, one joule is defined as the energy it would take to move an object a distance of one meter with a force of one newton. Alternatively, it is about the energy it takes to pick a bugger out of your nose.
The SI unit of mass, an amount that would weigh about 2.2 lb at sea level.
"The energy of motion." But wait. Motion is relative.
If you see a runner go by she will have some kinetic energy with respect to
you. But what if you try to "hit" on her and start running alongside? Then,
because she is motionless (at least on average) in your new frame of reference,
she will not appear to have any kinetic energy, whatever else she may have. The
formula for kinetic energy, K, of a mass m going a
speed v is
K = (1/2)mv2.
If the object of your
affections has a mass of 50 kg and a speed of
4 m/s, her kinetic energy will be 400 J, about
enough energy to warm up a cup of coffee a couple of degrees Fahrenheit (or a
whole guy by 20° F).
Now, once you start schmoozing along side her, her kinetic energy will be zero, but the earth will be going by at 4 m/s. That means the earth, with its mass of 5.97 × 1024 kg, now has a kinetic energy of 4.78 × 1025 J in your reference frame. This is enough energy to heat up about a thousand billion billion cups of coffee to boiling. I'll bet you had no idea you could generate so much energy by just trying to make time with a babe. Hey, why don't you tell her this. She will be impressed.
This is what the interaction between one surface sliding over another is called. (Physicists are therefore puzzled when the news talks about friction between two countries. Are their borders rubbing together? And, what if they have no common border? Physicists don't recognize the concept of "friction at a distance".) To a first approximation the force exerted by one surface on the other is proportional to the force pressing the surfaces together. This is the normal force. The constant of proportionality is called the coefficient of kinetic friction, μk, which is a property of two surfaces. (Glass slides over wood fairly easily, hence the coefficient of kinetic friction is relatively small compared to, for example, sandpaper sliding over wood.) See also static friction.
There's a saying at NASA right before a Space Shuttle mission, "Let's do launch." (From, of course, the phrase, "Let's do lunch." My response to this phrase is sort of like that of the older guy who was asked this by a younger colleague: "Son, I don't 'do' lunch, I eat lunch.") The launch angle for the shuttle is straight up, or 90° with respect to the ground. However, the shuttle is not a projectile, but is in powered flight. For projectiles, such as cannon balls and water balloons, the launch angle, αo, is defined as the angle between the horizontal and the direction in which the projectile is launched. Hence, for a projectile launched horizontally, αo = 0°. A projectile launched vertically upward would have αo = 90°.
(Now hear this! No Viagra jokes are allowed when discussing this topic!) This term refers to the relativistic effect where a moving object appears to grow shorter in the direction of its motion as seen by a "stationary" observer. The word "stationary" is in quotes, because it is a matter of perspective who is moving and who is not. This effect is only significant at speeds near that of light and is due to the requirement that the speed of light be measured the same in every inertial reference frame, which is a frame of reference that is not accelerating according to Einstein's special theory of relativity. This requirement means that the properties of space and time appear different for observers moving at different velocities. It is intimately connected to the concept of time dilation. It does not mean, however, that you can disappear from the universe, as claimed by the author of a certain highly error-filled book. He claimed you could travel in the x direction, shortening yourself in that direction as a result, then pick up speed in the y direction also, shortening yourself in the y direction as well, and finally speeding up in the z direction, turning yourself into a point. (So that's how those Klingon cloaking devices work.)
What Dr. Frankenstein said to the undertaker when he collected the body of his deceased wife. Also the effective distance through which a force operates in producing torque about an axis of rotation. It is defined as the distance from the rotational axis perpendicular to the line of action the force producing the torque.
You need a good line when going into a singles' bar if you want to get some action. How about, "Did you know that force is a vector, which can be symbolized by an elongated object such as an arrow, and the direction along which the force acts is called its line of action?" If that doesn't get you slapped silly, then perhaps you've found the woman you're looking for. Or a businesswoman (if you know what I mean). When you pick yourself up off the floor, bruised and bleeding, you can protest that there is a line of action associated with all vectors, not just force. For example, the line of action of velocity is the direction along which the motion takes place. Slap!! Of acceleration, the direction along which the motion is changing. Slap!! Slap!! Slap!! Male physics students should stay out of singles' bars.
This refers to acceleration for motion along a straight line. Average linear acceleration is the ratio of the change in velocity to the elapsed time over which the velocity changed, that is Δv / Δt. Since velocity is a vector, so is acceleration, meaning a direction must be specified for it. For straight line (linear) motion the direction is given by the direction of the change in the velocity. If an object slows down from 10 m/s to 6 m/s in 2 s, its change of velocity is 6 m/s - 10 m/s = -4 m/s, which is negative. Its average linear acceleration is therefore (-4 m/s) / (2 s) = -2 m/s/s (read as "minus two meters per second per second). Instantaneous linear acceleration is the change in velocity over a (practically unrealizable) infinitely small time interval (denoted as dv) divided by that time interval (dt). That is, a = dv/dt, which is called the derivative of v with respect to t in calculus. See also acceleration and tangential acceleration.
Known in politics as "Big Mo", desperately sought after by football teams, completely lacking in college students (especially toward the end of the semester), often referred to informally as "on a roll", this is one of the more important topics in physics. Simply put, linear momentum according to Newton is just the mass of an object times its velocity. You can also view linear momentum as the measure of the inertia of a moving object. Since the measure of inertia of an object at rest is its mass, whereas that of a moving object appears to be related to its mass times speed, it is not obvious what the connection between these two apparently separate views of inertia is. In fact, they appear to be incompatible. According to the idea that inertia is mass times speed, the momentum of a stationary object should be zero, which we know is not the case.
However, there is a connection: they are connected in relativity in a so-called "4-vector", a vector that has a time component as well as three spatial (x,y,z) components. This particular 4-vector is called the energy-momentum 4-vector. When an object is motionless, only the time component, which depends on the rest mass, is nonzero. (A popular interpretation of relativity theory says mass increases with speed.) When the object is moving, it has momentum as described above in addition to its rest mass. See also angular momentum.
This is about the tenth entry I've done where I feel like I have to say that we really don't know what this is, in this case "mass", and I'm getting damned sick and tired of it. I want one of those big-name physicists out there to make a definitive statement, but that probably won't happen anytime soon. Perhaps the discovery of the Higgs particle, if that occurs, will enlighten us as to what mass really is. Once again, we are limited to just being able to measure it, and this leads us to two types of mass. Inertial mass is measured through the property of inertia, which is a measure of the resistance of an object to a change in its motion. Gravitational mass is reflected in an object's weight, and is proportional to the magnitude of the force of gravity (weight) on the object. There is no reason, a priori to expect these two properties to measure out to the same amount of mass, but they do. Einstein's General Theory of Relativity provides a clue as to why this should be, but this is supposed to be a glossary, not an encylopedia, so I'm not going there. (However, see the discussion in Chapter Six.)
Like most other basic concepts in physics, "matter" is more of an unknown than a known. Physicists can measure things, and so we can describe properties of matter, such as size and mass, but exactly what it is eludes us. Defining matter, like I remember when I was in school, as "something that has weight and occupies space" only passes the buck to the concepts of weight and space. Anyway, an astronaut in the space shuttle is weightless but that doesn't mean he isn't made of matter. Matter has properties such as gravity and inertia that we can define (at least for purposes of measurement). We have discovered many of the elementary particles that make up matter. We also have found out that most matter is really just empty space, even though it appears solid to us. We have even learned that most of the matter of the universe is, so far, unknown and undetected, except through its gravitational effects.
Simply put, this is the sum of kinetic energy and potential energy. (This is also the complicated way to put it.) The mechanical energy of an object or system of objects will not change if the forces doing work on the system are all conservative forces. Nonconservative forces can either increase or decrease the mechanical energy of a system by doing work on it. Dissipative forces, such as friction, rob a system of its mechanical energy, turning it into, ultimately, heat energy.
The SI fundamental unit of length, somewhat over a yard in length. One meter equals 3.281 feet to four significant figures.
A parabola is an open curve given by the equation
y = ax2 + bx + c,
for a parabola whose axis lies parallel the the y direction. In elementary
physics, parabolas are the approximate paths of projectiles, ignoring air
resistance, and the ideal gravitational paths of small bodies with just enough
energy to escape a much larger body. The path of an ideal projectile is
actually an ellipse, but for motion near the Earth's
surface, it is so close to a parabola that there is no point in describing it
by the more difficult math associated with the ellipse.
The following list gives the metric prefixes, their symbols, and values.
Going small
(Let us bow our heads and remain still for a moment of
inertia.) I have no idea why the word "moment" is used
in this term. It has something to do with rotation, since
torque, for example, is often referred to as "moment of
force". (Let us bow our heads and shove each other for
moment of force.) The moment of inertia is the measure of the rotational
inertia of a nonrotating body, just like mass is the
measure of the translational inertia of a stationary body. Moment of inertia is
usually symbolized by "I", and is calculated by the formula
I = ∑miri2,
where the sum is over all the "point" masses (mi) that make up the
body times the distance from each point mass perpendicular to the axis of
rotation (ri) squared.
In one sense motion is an illusion. If you are standing on a curb, watching a car pass, you would say the car is moving. There is someone in the car, a young boy holding a tennis ball. The tennis ball is moving because it is in the hand of the boy in the car that is moving. However, to the boy the ball is not moving. He is holding it at rest in his hand. The boy now tosses the ball out of the car and backwards with just enough speed to compensate for the forward motion of the car. The ball was initially stationary with respect to the boy, but now he sees it moving backwards. You, however, see a stationary ball (after being thrown) that subsequently falls to the pavement.
From this example perhaps you can see that the motion that is observed depends on the motion of the observer. But who defines the motion of the observer? Another observer? Motion is therefore relative and some mistakenly believe this is the basis of Einstein's ideas of motion. The relative nature of motion was known well before Einstein, however. What Einstein added was the assertion that no matter what your motion is, when you measure the speed of light, you will always get the same value. In physics we describe motion with the concept of velocity.
More than one force may act on a body, but when you add them all up, their sum is called the "net force". Sometimes it is called the "total force", which is the same thing, that is, the sum of all the forces. Remember that forces are vectors, so forces must be added vectorially. That means, if the forces are not along the same line, they must be added using geometry. If the net force is nonzero, then the object experiencing this "unbalanced" force will accelerate in the direction of the unbalanced force.
The SI unit of force. It is a little less than a quarter of a pound (0.225 lb) and was named after those fig bars that used to come in packages this size. OK, you know that's bogus. It was really named after Isaac Newton, the famous English chef who invented the fig bars that carry his name. Not buying that either? All right, Isaac Newton was the famous English physicist who developed the concept of force as a mathematical formula, and the newton was named in his honor.
Here we are not talking about your average, God-fearing, tax-paying, family-raising force. Rather this is a force that presses two surfaces together. Actually, two forces are involved (as there always are, according to the third law of motion). Each surface presses on the other. For example, while sitting in a chair, the seat of the chair is pressing against your butt. In an equal and opposite fashion, you butt is pressing against the seat of the chair. These forces are actually called "normal", not because they are so ordinary (which they are) but because "normal" means perpendicular in mathematical lingo. These forces are always perpendicular to the surfaces they act between.
A type of periodic motion where the body or system moves back and forth over the same path. A good example is the Israeli-Palestinian conflict. If the oscillation dies out with time it is called a "damped" oscillation, like that in Northern Ireland (we hope). In physics an oscillation occurs when a body or system transfers energy back and forth between kinetic energy and potential energy. A damped oscillation is one where the energy of the oscillation is decreasing due to a force that dissipates mechanical energy, for example, friction.
An oscillation can be "underdamped", which means it loses energy slowly enough to oscillate at least once before dying out (coming to rest at the equilibrium position. An oscillation can be "overdamped", which means its position (and hence potential energy) changes so slowly (due to excessive friction, for example) that it takes a long time to die out, slowly approaching the equilibrium position without oscillating at all. (Hence, this is not really an oscillation per se, but would be if the dissipating force were reduced.) Finally, an oscillation can be "critically damped", where the dissipative force causes the oscillation to die out as quickly as possible. (This is how you want your car suspension to behave. For example, if you don't replace weak shocks, your car is underdamped and will bounce like crazy after you hit a deep pothole. Nevertheless your oscillation will cease after you lose control of the car and roll over. This gives rise to the expression, "damped if you do and damped if you don't".) Once again, this is not really an oscillation, but can be if the dissipative force is reduced.
Pascal was a small-time burgler who got in trouble with the Chicago mob. They fit him with some shoes that weren't exactly Nike Air and tossed him into Lake Michigan. That might be the end of the story, except that, on his way down, Pascal was struck by how the pressure seemed to increase over his body uniformly. "It's like the overlying water pressure is being distributed evenly throughout my body," he thought. He then concluded that the same must be true of the hydraulic system in the garage where he recently had his getaway car serviced. The mechanic applied compressed air to the hydraulic system. The pressure due to the compressed air was transmitted equally throughout the hydraulic fluid, causing a large force to be expressed on the piston that raised his car into the air for servicing. This was because force = pressure×area, and the area of the piston in contact with the hydraulic fluid was large - hence, so also was the force. "The pressure applied to a container is equally distributed throughout the fluid in the container and to its walls!" he concluded triumphantly.
As he sank through schools of puzzled fish, he vowed that if he were to get out of this alive, he would tell all the world about this principle he discovered. (As if the world at large, and that includes freshman physics students, could care.) However, luck was with him. Some police divers searching for Jimmy Hoffa came upon him and brought him to the surface. With his help, important mobsters were brought to justice, although he had to go into the witness protection program. To prevent him from being found, the feds gave him a new identity as a seventeenth century French scientist.
Periodic motion occurs when an object or system traverses the same path over and over again, always taking the same time (the period) to make a round trip. If the motion is back and forth over the same path it is an oscillation. However, the motion may also be in a circle or other closed loop. If you think your old Uncle Fred, who repeats stories over and over, is boring, take a look at Mother Nature and all her periodic motion. Be glad she doesn't show up at your family reunions.
Imagine a circular race track with a couple of cars moving around it in the same direction. You could describe the position of each of the cars by how far it was from the starting line. Alternatively, you could describe the angles the cars made from the starting line to their current positions along the circular track as measured from the center of the track. This would be a kind of phase angle. The difference in their angular positions would be the difference in the phase angles between the cars. If one car were at a position of 100° and the other, trailing behind, at 88° around the track, their phase angles would differ by 12°. If you use the lead car as the reference, this would be a -12° phase angle, since you have to go backwards (negative direction) to locate the trailing car.
Objects in periodic motion (of which circular motion is one form) can be located and related to each other by phase angles. Further, objects in simple harmonic motion can be related by phase angle, since simple harmonic motion is the back-and-forth projection of circular motion. Since periodic waves cause the particles the wave travels through to execute periodic motion, waves of the same wavelength and traveling in the same direction can be related to each other by phase angle.
The grandest and most exalted science of all. Those who practice it are the best and most important scientists, nay, human beings, in the world today. Besides this, it is the study of space, time, matter, and energy. If you are thinking of going into physics because "space, time, matter, and energy" sound cool, be warned that the profession of physics is filled with huge numbers of really intelligent people competing for a few low-paying jobs. (I said "intelligent" not "smart".) In other words, physics is like your spouse after twenty years of marriage: you have to love it for itself.
Physics students often ask, "What's the point!!" Well, actually, it could be you. Or, it could be any other object. If we are interested in the position of a particular object, we could paint a target on it and follow the center of the target as the position of the object. People do this, in effect, all the time. If you are on your cell phone talking to someone and they want to know where you are you don't respond, "My feet are on the pavement at Hollywood and Vine, my heart is about four and a half feet above my feet, my liver is around six inches below and to the right of my heart, my gizzard is ..., wait, I don't have a gizzard." Nobody wants to know where all your organs are unless you are a dead organ donor. So you treatyourself as a point. "I'm at Hollywood and Vine."
Of course, objects really are more than points and in physics we need to recognize this fact. It turns out that a convenient point with which to specify an object is the center of gravity of the object. This is the point at which the force of gravity appears to be operating. Another point is the center of mass, which, practically speaking, is almost always the same as the center of gravity. We can follow these points as if they were the position of the object we are tracking.
You may recall William Perry, "The Refrigerator", who played football with the Chicago Bears. He is credited for inventing the polar coordinate system. During the offseason, he was an arctic explorer, which is how he got his nickname, and gained initial fame for discovering the north pole. Anyway, once he and his team, the defensive unit of the Bears, reached the pole, they realized they had a big problem: Someone forgot the beer. Then they found out they had another major problem. Every direction was south. Home was south, but so was the Former Soviet Union, which was not so "former" at that time. They realized that with the wrong choice of south, they could end up cooling their heels in Siberia, or, worse, Fairbanks, Alaska.
William had an idea for determining direction. He chose one direction, the point where the sun rose, to be zero degrees. Then he measured an angle, called the azimuth angle, counterclockwise from that point as viewed from above. This defined a new type of coordinate system. (See the figure). The only problem with this solution was that the sun doesn't rise or set at the north pole during summer; it just circles around all day at the same elevation in the sky. Although they could not find their way home and had to be rescued by cameramen filming a National Geographic special, nevertheless polar coordinates caught on in the math community. One point is chosen for the origin of the coordinate system, and a direction is designated to be zero degrees for the azimuth angle. The direction to a point located in this coordinate system is given by the azimuth angle, θ, measured counterclockwise from zero degrees. The distance to the point is the radial coordinate, r, so that the point is located by the ordered pair (r, θ). Again, see the figure.
Position is the location of a point in a coordinate system, for example its x, y, and z coordinates in a 3-D cartesian coordinate system. Position is in reference to the origin of the coordinate system, so called because the coordinate axes emanate from that point.
A position can be what a politician says he, or she, believes in. A vector can be a pathogen. Therefore, in a perfect world, a position vector would be a disease contracted by politicians who adopt positions merely to get elected. Unfortunately, due to the existence of mathematics, the world is far from perfect, and we have to settle for the following state of affairs. A position, in the mathematical sense, is a point in a coordinate system, and a position vector is an "arrow" that extends from the origin of the coordinate system to the point in question. The magnitude of the position vector is its length, and the direction is given by angles (one angle in two dimensions and two in three dimensions).
"The energy of position." As in the case of kinetic energy, this energy is relative. But instead of being relative to an arbitrary velocity, as in the case of kinetic energy, it is relative to an arbitrary position. The details depend on what type of potential energy is involved. Each type of potential energy is associated with a force, a conservative force, that depends only on the relative positions of the bodies interacting with each other via that force. It is called "potential" energy, because it can be transformed into the energy of motion, kinetic energy. However, it is an energy in its own right, not just the potential of being energy, and resides in the field associated with that force. My wife does not believe me when I try to explain that I really am full of energy, even when I'm just watching baseball. She expects to see motion to prove there is energy there. Wives make lousy physicists.
A reader of my most excellent work of physics took great exception to my statement that the pound is not actually a unit of mass. He called me an uneducated so-and-so and demanded that I prove my assertion - that I drop everything and send him the references that would support my statement.
Sadly, I had to admit that I had been holding out. There is, in fact, an engineering unit called the "pound mass". This is not something that I feel freshmen physics and engineering students should be exposed to. They have grown up thinking the pound is a unit of weight. Personally, I think it is distressful enough to inform them that the pound is really a unit of force since weight is a force. You can't expect young people to be able to comprehend all the ugliness of the real world in one short semester. Some of these students will go on to be engineers. They will then have to deal with the unsettling existence of the pound mass, a unit that is no more than part of a dying effort to fend off the metric system. It pains me to tell you this, but if you use the pound mass in your equations, you must divide it by 32. ...Yes!! I admit it!! It's true! It's (sob!) true. The pound mass is only 1/32 of what it is said to be! I'm so sorry I had to be the one to tell you this. And it would not have happened had not some butthole brought it up.
Precision is not the same as accuracy. Precision has to do with reproducing a measurement close to the same value every time the measurement is taken. This does not guarantee the measurement is accurate. For example, you might weigh yourself on the bathroom scale a number of times with the result close to 150 pounds each time. However, if you are supporting yourself with one hand on the vanity, you would have a systematic bias (not to mention a dishonest bias) in your measurement.
This term usually refers to the type of stress known as "hydrostatic" pressure (due to the common experience of feeling squeezed when under water) or "confining" pressure. Pressure is simply defined as the force exerted per unit area.
There are more units of pressure than you can shake a tire gauge at. It seems every application has its own unit. The SI unit of pressure is the pascal (newton per square meter, symbolized as "Pa"). This is a small unit of pressure. It takes over 100 000 Pa to equal the pressure of the atmosphere at sea level. Meteorologists have traditionally used inches of mercury, derived from the use of the mercury barometer. A pressure of one inch of mercury is the pressure you would feel if you were swimming in a pool of mercury one inch below the surface. (Actually, mercury is so dense you could not possibly get below the surface unless weighted down with gold ballast.) Since then, the "whethermen" (as my brother calls them) have gone to millibars, which is the same thing as hectopascals. Then there are pounds per square inch (psi), centimeters of water (used in medical applications), torr (the same as centimeters of mercury), dynes per square centimeter, the standard atmosphere, ... the list goes on and on.
Pressure measured with respect to the vacuum is called absolute or total pressure. In a vacuum there can be no pressure - at least not from ordinary matter. When dealing with such things as tires and tanks, you encounter what is called gauge pressure. Gauge pressure is the pressure in a container measured with respect to the prevailing (ambient) pressure outside the container. A tire that has 35 psi of pressure as measured by a tire gauge actually has 49.7 psi total pressure if the tire finds itself in a prevailing pressure of one standard atmosphere. The 35 psi is measured with respect to the atmosphere, not outer space. A flat tire therefore has a pressure of 14.7 psi absolute, even though its gauge pressure is zero.
Gauge pressure can be negative. A vacuum cleaner creates a negative gauge pressure. There was a short ad campaign (foreign-produced) many years ago with the slogan, "Nothing sucks like an Electrolux." Consumers rolled in laughter at this. Being quite physics savvy, they knew that suction is not real. Vacuum cleaners, for example, don't suck dirt. The low pressure created by the machine's fan produces a pressure difference, and air flows from high to low pressure. The Electrolux ad campaign was thus exposed to glaring ridicule and was quickly withdrawn. They should have used a different slogan such as, "You may think our product sucks, but that is only due to your ignorance of physics."
Actual title: Naturalis Philosophiae Principia Mathematica, which is Latin for, "Up Yours, Robert Hooke". This was Isaac Newton's ground-breaking work on physics, which Edmund Halley finally persuaded him to write. It is so important to the history of physics that I blush to admit I had not read any of it - until, that is, I read where a noted feminist philosopher had referred to it as "Newton's Rape Manual". Now that got my attention. Sexandviolence! I eagerly got a copy and began reading it from cover to cover. Sadly, however, there was nothing from one cover to the other but physics, physics, and more physics. As much as I hate to say it, although it may be a great work of physics, as a rape manual it is pretty much a total loss.
This is an object that is launched ("projected") with a force of short duration and thereafter travels without propulsion or significant aerodynamic support (such as a glider has), like a spit wad or a water balloon. In other words, the most significant force acting on the object, until it hits the teacher, is gravity. Its path is a parabola in the absence of air and a distorted parabola in the presence of air resistance. The greater the air resistance the greater the distortion.
The radian is used to measure angles, just like degrees, except that the radian is a natural unit of angular measurement. Practically speaking, this means you don't have to supply a conversion factor in numerous equations where angles appear if your angular measurement is in radians. Imagine a circle of radius r. It could be a pie. In fact, since I'm writing this with Thanksgiving coming up in a few days, it could be a pumpkin pie. With whipped cream on top! Yum! Imagine you slice yourself a piece of this pie. The arc length of the crust, s, divided by the radius of the pie, r, is the angle of your piece of pie in radians. That is, θ = s/r. Since mathematicians have defined the number "π" such that a complete circle has a circumference of 2πr, it follows that 360° = 2πr / r = 2π radians. Then 180° = π radians, 90° = π/2, 45° = π/4, 60° = π/3, 30° = π/6, etc. The only mystery here is why 360°, which is one complete pie, equals 2π. Obviously, the stupid mathematicians really screwed up, because 360° is only one pie! By claiming one pie is really two, I think the mathematicians are trying to claim more than their share of the pie. As a result we are unfortunately stuck with this absurd definition of the number pi.
If you are talking about physics and not a beach party on South Padre Island
during spring break, the short answer is the following equation,
K = (I / m)1/2
where K is the radius of gyration, I is the moment of
inertia of a body about its axis of rotation, and m is its mass.
Geometrically, the radius of gyration of a body is equal to the radius of a
hoop of the same mass with the same moment of inertia when rotated about its
symmetry axis. Since a hoop of radius R
rotated about its symmetry axis has a larger moment of inertia than any other
shape of the same mass distributed no farther than R from its rotational axis,
K is larger the more distantly the mass of a body is distributed from the
rotation axis. For a hoop rotating about its
symmetry axis, K = R, its largest possible value, whereas a rod rotating about
an axis along its length has the smallest radius of gyration. What
about the radius of gyration at that beach party? Sounds like a job for
further, hands-on, investigation.
These are errors that tend to fall with equal probability on either side of the actual value. Therefore the more measurements you have, the more the negative errors will tend to cancel the positive ones. The total error resulting from random errors can be lessened by increasing the number of measurements. Just think of what this means. If only our government could make lots of errors on both sides of a policy, eventually they might get it right.
This term is used to describe how far a projectile goes when it is launched on a horizontal surface (at least in freshman physics). Therefore, the distance traveled by the projectile when it reaches the same level it had when it was launched (strikes the ground) is the range. It is a little known fact that, during the Texas Revolution, artillerymen had limited training facilities. To sharpen their skills against a mobile enemy, they would conduct target practice by trying to hit cattle roaming on the prairie. This is where the term "range cattle" comes from.
Motorists traveling rural Texas highways aren't the only ones who have to put up with annoying speed zone changes. Waves have to also, because as they propagate along, minding their own business, the medium through which they propagate may change its propagation characteristics, causing the waves to either speed up or slow down. This phenomenon is called refraction. If a wavefront runs into a situation where the wave speed in the medium varies from faster to slower across the wavefront, it bends toward the slower side, causing a change in the direction of propagation of the wave. If the left side of a wavefront is slower than the right side, for example, the direction of propagation of the wave will slant toward the left. This is especially true if the wave propagates into the "liberal media". However, this circumstance appears to happening less and less. For example, waves propagating into FOX News tend to emerge in the opposite direction. This is not really refractionary so much as it is reactionary.
We all know that a "resonant voice" is an asset to an orator. Actually, however, the term "resonant" here has a different meaning than in physics. What is referred to here is a loud voice. The louder your voice, the more weight your words carry. Actually, however, the term "weight" used here has a different meaning than in physics. What is referred to here are compelling words. The more compelling your words, the more your audience is moved. Actually, however, the term "moved" here has a different meaning... OK, it looks like I'll never get to the subject at hand if I keep this up.
Resonance can occur when the frequencies of two coupled systems or components of a system are nearly the same or are close to multiples of each other. For example, if your bring a tuning fork vibrating with a frequency of, say, 256 Hz (Hz = hertz = oscillations per second) close to a tube in which the air can vibrate at 256 Hz, the sound emitted by the fork will cause the air in the tube to start vibrating. What you will hear is an increase in the volume of the sound. This will also happen if the tube can vibrate at 512 Hz, since 2×256 = 512. If the systems can exhibit chaotic behavior, a resonance can cause a marked change in behavior. For example, if an asteroid between Jupiter and Mars has an orbital period one-half that of Jupiter, it will get an extra gravitational tug from Jupiter every other orbit, at the same point in space. The result of these tugs (called a 2:1 resonance) will be to knock the asteroid out of its orbit. This is why you don't find asteroids between Mars and Jupiter with this orbital period. Instead three is an "asteroid gap" in the vicinity of that orbit.
One version of this is the fact that everything is made for right-handed people. Left-handed people are "sinister" (from the Latin word for "left") and not be trusted or given an appropriate table setting. For example, in baseball you have right-handed pitchers and "crafty" left handers. You never hear right handers being described as "crafty". Instead, you may hear announcers use the phrase, "He's a hard-throwing right hander." Honest. Dependable. A paragon of decency, not like those crafty lefties.
In physics this highly justified suspicion of left-handed people is embodied in its version of the right-hand rule. Actually, there are different variations of this rule, but they all hark back to the definition of the cross product. A click on the link in the preceding sentence will describe the "canonical" definition of the right-hand rule.
A fundamental variation is used to define right-handed cartesian coordinate systems. Stick your right hand out and shake it all about. Do the Hokey Pokey and... Sorry, got carried away. Stick out your right hand and extend your thumb, index finger, and middle finger at right angles to each other. Your thumb is the x axis, your index finger the y axis, and your middle finger the up-yours axis (also known as the z axis).
Another important variation has to do with rotation. Curl the fingers of your right hand in the direction of the rotational property (e.g., angular velocity), and your thumb points in the direction of the (axial) vector that corresponds to that property. (Check out Figure 8.12.)
You will come across the variations of the right-hand rule as you read my "textbook" (or most college physics texts). In electromagnetic theory there are so many right-hand rules, physics students have actually gone insane and gnawed off their right arms. A listing of these can be found here*. Read at your own risk.
A rigid body in physics is one where every atom is always in the same position with respect to every other atom in the body, even when outside forces are applied to the body. In other words the body cannot be deformed - its shape is always exactly the same. Of course, no real body, with the possible exception of Al Gore, is perfectly rigid, but it makes a good approximation in physics, since many objects are only minutely deformed by ordinary external forces. Objects made of wood, concrete, steel, etc., make passable rigid bodies, because they resist external influence. Congressmen, on the other hand, make really crumby rigid bodies.
Isaac Newton's nemesis. Brilliant, conniving, and evil, he was to Isaac what Professor Moriarity was to Sherlock Holmes. Bent on destroying the world, or Newton's reputation failing that, he was thwarted time and again by the superheroics of Professor Newton. At least this is my fantasy. Apparently there was bad blood between the two scientists. Robert Hooke is mainly remembered for "Hooke's Law", an expression of the properties of elasticity. I understand he did other important stuff, but, hey, if they don't name it after you, nobody remembers or cares.
This lack of recognition is, in reality, extremely unfair. Robert Hooke, despite the dispute over whether he helped invent the theory of gravity and his boasting about it (which irritated Edmond Halley enough that he persuaded Newton to publish, just to shut Robert up), was a scientist in the class of Newton. He was especially brilliant in microscopic work and coined the word "cell" to describe that biological building block. Apparently, he was difficult to get along with and raised a stink about Newton's work on optics when Newton was elected to the Royal Society, something Newton apparently never forgave. However, Newton was also an egomaniac. Lots of scientists are egomaniacs... and arrogant. If you don't believe me, attend one of my classes.
A scalar is a quantity that can be placed on a linear scale, such as length, temperature, time, etc. It has units and a value and that's it.
The fundamental unit of time in SI units. You all should know what a second is - the attention span of the average physics student.
Since Galileo came up with the first law of motion, this is not really Newton's second law. Also, since it is apparently the definition of force, it is presumably not a law in the sense of what is usually meant by a law of nature. So maybe it should be called "Newton's one and only definition of force - no pictures or accounts of which can be reproduced without express written permission of the NFL (Newton's Force Lawyers)".
However, it turns out to be much more than a mere definition. It hypothesizes that there exist natural interactions that can cause acceleration. What it says is, "A force is that which, if exerted on an inertial mass unopposed and unaided by any other force, will cause the mass to undergo an acceleration. The magnitude of the force is equal to the mass times the magnitude of the acceleration. The direction of the force is the same as the direction of the acceleration." Hence causes (forces) are postulated to exist which produce effects (accelerations). Now we have to look around the universe and see if there are any phenomena that correspond to this hypothesis. Indeed there are, and these forces constitute the basis of classical physics.
Stands for "Systeme International", the international system of units, which forms the basis for other recognized units, such as those of the BS. (I know what you're thinking! For your information BS stands for British System of units, not that other thing! ...Come to think of it, you may have been right in the the first place.) The SI is a metric system, whose fundamental units for mechancis are the meter, the kilogram, and the second. Derived units include the radian, newton, the joule, the watt, the hertz, and the pascal.
The only digits in a number that are meaningful. The rightmost significant figure is called the "least significant figure" and represents the best estimate of the precision. There is disagreement about the meaning of the most significant figure. Mathematicians claim it is the leftmost digit; however, some say it is Christy Brinkley and others say it is the net worth of Bill Gates.
Take a rock, tie it to a string, and twirl it in a horizontal circle at a constant rate. Do this at sunrise or sunset next to a wall such that the stone's shadow is projected on the wall by the sun shining brightly from a clear horizon. (Anywhere on the High Plains of Texas would be ideal, where, as the residents say, you can see farther and see less than anywhere else on earth.) The shadow the rock makes on the wall will go back and forth in simple harmonic motion.
This motion is called "harmonic" because, if you were to move a pen back and forth over paper in such a way as to mimic the motion of the rock's shadow on the wall while, at the same time, the paper underneath the pen was pulled at a constant speed at right angles to the motion of the pen, you would trace a harmonic (that is to say sinusoidal) curve. (See the figure.) A system that oscillates in simple harmonic motion is called a simple harmonic oscillator.
There is some important terminology associated with simple harmonic motion. The time it takes for one oscillation is called the period. The number of oscillations per unit time (= one divided by the period) is the frequency. The maximum displacement is called the amplitude.
Now, why the term "simple" is included in the term "simple harmonic motion" is not clear to me. The motion, when analyzed in detail, is definitely not all that simple. Furthermore, I have never seen the phrase, "complicated harmonic motion". Frankly, I think the word "simple" was diabolically added to lull physics students into a false sense of security when this subject matter is covered. Wish I had thought of it.
Another theory has it that the word "simple" wasn't really added by physics instructors who went over to the dark side of the restoring force. An elastic medium can vibrate at many different frequencies at once when a wave passes through it. (For example, sound waves passing through the air.) Although the vibration of each frequency is simple harmonic, the combined vibration, though harmonic, is quite complex.
An oscillator that oscillates in simple harmonic motion is a simple harmonic oscillator. A simple harmonic oscillator has the property that the magnitude of its restoring force is proportional to the displacement of the system from equilibrium. (When this is the case, the force is said to be elastic.) The restoring force is therefore greatest when the system is farthest from equilibrium. This also means, from Newton's second law of motion, the acceleration towards equilibrium is greatest when the system is farthest from equilibrium. The farthest distance from equilibrium is called the amplitude of the motion.
Since the system is always accelerating towards equilibrium (either speeding up while moving toward equilibrium or slowing down while moving away from equilibrium), it has its greatest speed when it moves through its equilibrium position. On the other hand, it comes to a momentary rest when at its greatest displacement from equilibrium. These are called the "turning points". (Currently, as I write, Iraq is about as far from equilibrium as you can get, so maybe we are at a turning point there. However, this would only be true if Iraq is a simple harmonic oscillator.)
Energy-wise, a simple harmonic oscillator can be viewed as a system where there is an internal exchange of kinetic and potential energy. That is, as the oscillator speeds up, potential energy is being converted to kinetic energy. As the system slows down, potential energy is being stored at the expense of kinetic energy. The type of potential energy depends on the type of oscillator. As examples, a mass-spring system has elastic potential energy, whereas a pendulum has gravitational potential energy.
In the world of graphs, slope refers to the rate of climb or descent of the curve on the graph. So, it is just like the slope you are familiar with if you are a skier. There are your bunny slopes and your expert slopes. The measure of slope is "rise over run", that is, vertical change (rise) divided by how far horizontally you went to get that vertical change (run). Now, "rise" does not necessarily refer to an increasing altitude; a negative "rise" is actually a fall. Thus for a positive slope you are either trudging uphill (if you are a cross-country skier) or taking the ski lift (if you are a downhill skier). A negative slope means you are effortlessly swooshing downhill, passing trees and slower skiers (especially the ones embedded in the trees). The steeper the "slope", the faster you zip downhill and the more spectacular your wipeout.
For a straight line the slope is the same at all points, because the ratio rise to run is the same no matter what two points on the graph you choose for the computation of the slope. What if you have a curve rather than a straight line? To compute the slope at a given point on the curve, you have to draw a straight line tangent to the curve at that point. This line will have the same direction as the curve at the point where it is drawn. The slope of this line is then the same as the slope of the curve at that point.
Tiny angles are great as far as engineers and physicists are concerned,
because they are both useful and easy to handle. In fact, if it weren't
for tiny angles, scientists would likely be writing up horoscopes; engineers
would probably have the arch, but couldn't figure out how it works;
and physicians would still be bleeding their patients. (Actually, this
practice continues today, but the blood is green. If my personal physician
reads this: Just kidding!)
This is because tiny angles allow us to make analyses that would not be
possible otherwise. For example, it allows us to find the magnitude for
the centripetal acceleration,
ac = v2 / r.
The small angle approximation arises because, for very tiny pie slices cut
from circles, the pie slice is almost a triangle with the smallest side
corresponding to the arc, Δs, subtended (cut) by the two radii, r,
of the slice. (See the figure.)
This arc is nearly the same length as the corresponding chord, Δl.
Since the arc length divided by the circle's radius is the definition of the
angle, Δθ, of the pie slice measured in radians, this angle is
approximately given by
Δθ = Δs / r ≅ Δl / r.
This relationship gives rise to the following closely related approximations.
sinθ ≅ θ, where θ << 1 radian.
tanθ ≅ θ, where θ << 1 radian.
Note that the angle must be measured in radians for the above
approximations
to hold. In the limit where θ gets smaller and smaller ("goes to zero"
in calculus terminology), the small angle approximation becomes an equation
involving "infinitesimals".
dθ = dl / r.
Space! The final frontier! It's big! Really, really, really big!! OK, so no one knows what space is exactly, but we have discovered some of its properties. We can define coordinate systems and locate points in space. We know that there is a property called distance that can be defined between points. We know that we can only define three perpendicular coordinate axes (as far as we can tell), and we say we live in three dimensions. In relativity theory space is an aspect of a larger entity called spacetime, reflecting the intimate connection between the spatial directions and time.
I've heard that some have claimed that time is just another dimension of space. It's easy to demonstrate this is not true. Walk to your front door. Now, facing the same direction, walk back to where you started. You just moved backward in space. That was easy to do, but how would you move backward in time such that you arrived where you started when you started? If time were merely another dimension of space, then it would be as easy to go back in time as it is to go back in space.
In the special theory of relativity, space and time are a composite property of the universe. How an observer views space and time depends on his motion through spacetime. Space and time for observers moving with different velocities are described by something called the "Lorentz transformation", which allows you to calculate what distances and times are for one observer, knowing what they are for another. In general relativity spacetime becomes more complex, as it can be curved, or "warped", by the presence of matter. In the case of space warping this means, for example, that in warped space you might measure the angles of a triangle and find they didn't sum to 180°.
In relativity theory space and time are combined in a mathematical relationship called a "metric". In this relationship the algebraic variable for time has the opposite sign as those for space. That is, if the space variables have negative signs, the time variable has a positive sign. This is of fundamental importance and the reason time progresses in a way that space does not. For example, the reason you would inevitably move toward the center of a black hole, should you be so unfortunate as to be swallowed by one, is because space and time are so warped in a black hole that the radial coordinate inside the hole takes on aspects of time. And, as time must go forward, so must you move toward the hole's center and your certain destruction.
In spite of our ability to describe spacetime mathematically and make measurements, no one knows what it is, exactly, or why it even exists. Why three dimensions? Why not only two? Why not 666? Some theories postulate there are extra dimensions of space that we haven't yet detected because they are too small to move into. Yet only one dimension of time is thought to exist.
Speed is the rate at which distance is being covered. We are most familiar with speed expressed in miles per hour, due to our driving experience. In physics usage, speed does not indicate which direction the motion is in. For that, physics uses the concept of velocity
If two surfaces are pressed together (see normal force), and a force is exerted to slide one surface over the other, there will be a resistance force called the force of friction. When the surfaces are not sliding, this force is called the force of static friction. The force of static friction will equal the force exerted to cause the sliding until the applied force reaches a magnitude that static friction can no longer resist. This is where the surfaces are on the verge of sliding, and this force is proportional to the normal force pressing the surfaces together. The constant of proportionality is called the coefficient of static friction, μs. After sliding commences, the resistance force will be that of kinetic friction.
Strain is the deformation of matter that results from stress . If a body experiences uniform compressive stress on all sides (hydrostatic pressure), it will maintain its shape (if it is uniform in makeup). In general, however, strain means a deformation of the size and shape of a body or collection of matter. The easiest strain to understand is elastic strain, where the deformation is proportional to the stress applied. (Double the stress, double the deformation.) The elastic response of substances to stress is characterized by elastic moduli, which are analogous to the force constant of a spring. The larger the elastic constant, the more the substance resists deformation just as the greater the force constant of a spring, the stiffer the spring is. Important elastic moduli are the bulk modulus, which characterizes how difficult it is to compress a substance; Young's modulus, which characterizes how difficult it is to stretch a substance; and the shear modulus, which characterizes how difficult it is to distort a substance's shape by shear stress.
If you feel bent out of shape, then you should blame stress, not your physics professor, because that's what stress does. It deforms matter. Stress is the force per unit area acting on a surface. The surface may be an actual one between two different substances or an imaginary one like the distinction between the U. S. Constitution and the Supreme Court during Presidential elections.
A physical example of an imaginary surface would be the equatorial plane, which is not a real surface but is envisioned to separate the earth into northern and southern hemispheres. Imaginary planes are used to analyze internal stresses in bodies. You imagine a plane to exist inside a body and analyze the stresses that would be experienced by that plane if it existed.
There are two distinct ways force can be applied to a surface: perpendicular or parallel to it. (In general, of course, the force is applied at an angle but can be resolved into parallel and perpendicular components.) Stress caused by forces perpendicular to a surface produces either compressive stress, if the force is directed toward the body or tensile stess, if directed away. Stress caused by forces parallel to a surface produces a shearing effect and so is called shear stress. This all sounds fairly simple, except that the number of combinations of forces and surfaces can be large. A cube, for example, has six surfaces, and each can be subject to a combination of shear plus either compressive or tensile stress (or no stress), leading to 48 possible combinations in all. Stress can change either the size or shape or both of a body or collection of matter. This deformation is called strain.
This is an error that throws off a measurement's value the same way each time the measurement is taken. These errors do not tend to cancel out when the average is taken. However, once a systematic error has been discovered, and if it was the same throughout the measurement process, it can be corrected. That is, unless it is a systematic error made by a bureaucracy in its dealings with you. In that case an error was not really made, but if it was, it was your fault.
Tangential acceleration is the same as linear acceleration in the following sense. Linear acceleration is rate of change of the velocity of an object in straight-line motion. It is the same as rate of change of the speed of the object but with a plus or minus sign. The plus sign corresponds to the velocity becoming more positive (or less negative), and the negative sign to the velocity becoming less positive (or more negative).
This sign convention is also true for tangential acceleration, except that tangential acceleration can occur along a curved line as well as a straight one. One direction along the curve is designated positive and the opposite direction negative. If the object is either speeding up in the positive direction or slowing down in the negative direction, the tangential acceleration is positive. If the object is either slowing down in the positive direction or speeding up in the negative direction the tangential acceleration is negative. An object traveling in a curved path has centripetal acceleration in addition to tangential acceleration.
Tangential and centripetal acceleration are illustrated in this figure, which shows a particle moving in a circle and speeding up. The figure shows the velocity changing from v1 to a faster velocity, v2. Note that the change in velocity can be resolved (divided) into a component in the centripetal direction, Δvc, and a tangential component, vt. The figure shows a "finite" (that is, not an infinitesimal, or ultimately tiny) change in velocity; for such a finite change this division is approximate.
The instantaneous change in velocity with respect to time (total acceleration) can be exactly resolved into the instantaneous centripetal change with respect to time (centripetal acceleration) and the instantaneous tangential change with respect to time (tangential acceleration). The centripetal and tangential accelerations are perpendicular to each other such that they form two sides of a right triangle with the hypoteneuse being the total acceleration as shown in the figure.
Let's see. Newton's first law of motion was really due to Galileo, his second law is a definition (and, actually, much more) and not exactly a law and therefore neither first nor second, so the third law of motion must really be his first and only law of motion. It might appear, therefore that Newton didn't do as much as claimed, but, in fact, he was the one who put physics on a modern footing. Theoretical physics can be thought of as beginning with the efforts of Sir Isaac, just as experimental physics began with Galileo.
His "third law" expresses the idea that something cannot act on something else without being acted upon in turn. To whit: "If body A exerts a force on body B, then body B exerts an equal and opposite force on body A." These two equal and opposite forces are called an "action-reaction pair". Each member force of this pair is called an "action-reaction partner". Note that these forces must act on separate objects. A pair of forces acting on the same object is not an action-reaction pair. Also the action-reaction partners must be the same kind of force (for example, both gravitational).
"Does anyone really know what time it is? Does anyone really care?" The old song could have asked a more profound question, "Does anyone really know what time is?" Answer: "No". Time has properties or we couldn't measure it. Or perhaps it isn't really time that has properties but just sequence of events. But isn't that what time is? A sequence of events? But then don't you need the concept of time to define what an event is? Whew! Well, what we do is make a device that generates a series of "events" and say that measures time.
Physicists look at time a little differently depending on what type of physics you are talking about. There is the time of thermodynamics, the time of relativity, the time of classical mechanics, etc. No one understands time. Hey, if you can just tell time, then be content and quit bugging me about it.
This could be a painting by Salvador Dali but isn't. It's weirder than anything Sal could come up with. As someone goes faster and faster, close to the speed of light, his time appears to be running more and more slowly as viewed by others "at rest". This gives rise to the famous "twin paradox", where one identical twin takes off in a space ship traveling at almost the speed of light while the other remains behind. The traveling twin, who has aged only a few weeks, returns to find his stationary twin is now an old man. However, since the twin left behind appears to be moving from the point of view of the traveling twin, and so is observed by the traveling twin in dilated time also, how come it isn't the traveling twin who is old? The answer is that the traveling twin made a round trip, which means he had to accelerate at some point (slow down, turn around, and speed up). But an accelerating reference frame is not an inertial reference frame, and this is where the difference between the two twins comes from. For more details you need to consult a book on Special Relativity.
You can think of torque as being a sort of "off-center" force. Say you have a merry-go-round on a wheeled pallet. If you try to push the pallet forward by pushing on the merry-go-round, you had better push exactly toward the merry-go-round's center. Otherwise, you will end up mostly turning the merry-go-round and not moving the pallet forward. What you want to do is to not create a torque about the center of the merry-go-round. If you grab hold of the edge of the merry-go-round and push in a direction at a right angle to its center, you will be applying a maximum torque. The merry-go-round will begin to rotate, but little, if any, motion will occur for the pallet itself. To optimize the forward motion of the pallet, you need to push toward the center of the merry-go-round to move the pallet forward. This would produce zero torque. Hence torque about an axis of rotation is created by pushing along a line that is off-center from the axis of rotation. In physics, the amount of torque is calculated by multiplying the applied force by the perpendicular distance from the axis of rotation to the line of action of the applied force. This distance is called the "lever arm". Hence, torque is force times lever arm.
If you've traveled much by air, you are familiar with this type of motion,
encountered in holding patterns around airports. An object in uniform circular
motion travels in a circle at constant speed. Although its
speed is constant, it is nevertheless changing its direction of motion and hence
its velocity. This change of the direction of motion
results in centripetal acceleration, an acceleration
that is directed toward the center of the circle of motion. Therefore, as an
air traveler you can take comfort that while in the holding pattern you are
constantly accelerating toward the airport. Try not to think about the fact
that, in spite of this, you are not getting any closer. The formula for the
magnitude of this acceleration is
ac = v2 / r.
A vector is a quantity that has units and a numerical value plus a direction. Quantities like force and displacement have directions as well as magnitudes associated with them and are vectors. The magnitude of a vector is always a positive number and represents how big the vector is. A vector's direction can be described by angles (one angle in two dimensions and two angles in three dimensions). A vector can also be thought of as an ordered set of quantities. In two dimensions there are two of these, in three dimensions three - one quantity for each perpendicular direction (x, y, and z axes). These are called the components of the vector. In one dimension a vector consists of one quantity, and whether it has a plus or a minus sign determines whether it points in the positive or negative direction. A one-dimensional vector is mathematically, but not conceptually, identical to a scalar that can take positive and negative values. For example, a velocity of +20 m/s points in the opposite direction to a velocity of -20 m/s, but the same is not true for +20° C vs -20° C.
A vector can be modeled as an arrow. Think of a vector as an arrow in a 3-D Cartesian coordinate system, with its tail at the origin. (See the figure). Conceptually, you can find its components like this. You proceed along the x axis (in either the positive or negative direction, as appropriate) until your line of sight to the tip of the arrowhead is perpendicular to this axis. (See the figure again.) Then you walk in the (+ or -) y direction until your line of sight to the tip is perpendicular to the y direction. (It will also be perpendicular to the x direction at this point.) Finally, you find you can walk in the (+/-) z direction directly to the tip of the arrow. The three straight lines that made up your stroll are just the x, y, and z coordinates of the vector. (Admittedly, you would have to have special powers to take a walk of this sort.)
Another way to look at components is that these are the vectors lying along the x, y, and z directions that add, vectorially, to the vector in question. Or, if you are a trig freak, take the cosine of the angle between the arrow and the x axis and multiply it times the length (magnitude) of the arrow. This gives you the x component. Do the same for the y and z axes, that is use the cosines of the angles between the arrow and those axes, to find the y and z components. Note that, with this method, if the angle is greater than 90°, the component will be negative (as it should be for such an angle).
This technique can be generalized to find "how much" of a vector lies along any given line: Find the angle between the vector and the direction chosen. Multiply the magnitude of the vector by the cosine of this angle. The result is the component of the vector in that direction.
Finding the components of a vector is called "resolving" the vector into its components. Replacing the vector with its components (which are equivalent to the original vector, since the components add up to the vector) is called "decomposing" the vector. Finally, note that you can choose your x,y,z axes any way you please (although they have to be at right angles and satisfy the right-hand rule to constitute a right-handed "cartesian" system), so the components you get depend on your choice of axes.
Velocity is speed plus direction. Speed is the rate at which distance is being covered, but has no reference to direction. Like the old airline joke: "This is your captain speaking. I have good news and bad news. The good news is that we are making excellent time. The bad news is I have no idea which way we are going." Velocity is a vector, meaning it must have a direction as well as a magnitude. A airplane flying north at 200 mph would have the same speed as one traveling east at 200 mph; however, they would have different velocities.
Graphically speaking for motion in a straight line, velocity is the slope of the position versus time plot for an object. To find the velocity of the object at a certain position (generically designated as "x"), draw a straight line tangent to the x versus t curve at the point corresponding to that position. Then find the slope of that line, which will be the (instantaneous) velocity. See the figure, which shows a negative velocity at time t1 and a positive velocity at time t2 (moving in the positive and negative directions, respectively).
Stress has the units of force per unit area, velocity distance over time, and d is a distance. Therefore you deduce that viscosity has the units of force per unit area times time, Pa·s (pascal-second) in SI (1 Pa = 1 N/m2). However, the common unit of viscosity is the poise, which is one-tenth as large as a Pa·s.
The SI unit of power, the rate of expenditure of energy, equalling the expenditure of one joule of energy each second. Named after James Watt, who didn't actually invent the steam engine, nor did he claim to have invented the steam engine. Instead, he improved steam engine technology and ran a series of TV ads that went something like this: "James Watt. I don't make the steam engines you buy; I make the steam engines you buy better."
A wave is a vibration of a medium that moves (propagates) from one point to another, carrying energy, but not the medium itself, with it. A wave pulse is a vibration that passes through a given point in the medium in a relatively short period of time. A periodic wave is a wave that repeats itself (both in space and time - think of a boat rising up and down on an "endless" sea of ocean swells). A wavetrain is a periodic wave or group of periodic waves of finite extent. (All periodic waves belong to wavetrains, since no wave goes on forever.) The time over which a periodic wave repeats is its period. Dividing the period into one (taking its reciprocal) calculates the frequency of the wave. The distance between adjacent equivalent points on a wave (for example, the horizontal distance from crest to crest of ocean swells) is called a wavelength. A "sinusoidal" wave (also known as a harmonic wave) is a periodic wave whose profile (both in space and time) looks like a sine function. (The cross section of an ocean swell would make an approximation to this shape.)
Waves are disturbances that move from one point to the next. A disturbance in one part of a medium carrying a wave creates new disturbances in its vicinity which result in the motion of the wave. Therefore waves are said to propagate and the speed of their motion is the propagation speed. For periodic waves this speed is the wavelength divided by the period (as in speed = distance / time). The amplitude of a sinusoidal wave is defined as the magnitude of the disturbance of the medium from its undisturbed level. (For an ocean swell, this would be the distance from the level of the calm ocean either up to the crest of a swell or down to the bottom of a trough. This should not be confused with peak-to-peak amplitude which is the magnitude of the disurbance from the bottom of a trough to the top of a crest and is twice the actual amplitude if the wave is harmonic.
Something most people in America are trying to lose, including me. Now the
weight of an object is the force of
gravity on that object, given by the formula
W = mg,
where m is the object's mass and g is the local
acceleration of gravity, so one strategy
we phatty physicists have tried, without success, is to reduce the pull of
gravity. That could be accomplished by reducing the gravitational constant, G.
(Gackkkk!!, there's that word reduce again!) But just like the extra
weight on your body, it is no easier, unfortunately, to reduce G. Physicists
have pretty much come to the conclusion that nothing can be reduced. It's
somehow related to the law of entropy
which says the universe tends toward a state of greater disorder. (This
disorder was apparently introduced into the universe with the
invention of children.) So, the universe is falling apart and expanding
at the same time. Kind of like us middle-aged people.
Anyway, see the related subjects
apparent weight and weightlessness.
When your apparent weight is zero, you are experiencing "weightlessness". This does not mean that you are actually weightless (hence the quotation marks), because that would mean the force of gravity acting on you would be zero. Consider the dude in the apparent weight discussion whose bathroom is in an elevator. Now, if the worst should happen and the elevator cable broke, at least he would be in a good place to be scared <excrement>less. Assume he is on his bathroom scale when this happens. Without the support of the scale, which was supported by the elevator floor, which was supported indirectly by the cable, the scale cannot exert a force on him. The scale reading is zero. At first he thinks his diet is working, maybe a little too well. Then he realizes the awful truth - that it is going to cost him a bundle to fix his plumbing. His actual weight is not zero - this is the force that is causing him to fall. However, the scale reads zero, and that is his apparent weight. A similar thing happens to astronauts in orbit. Their actual weight is not zero, but, being in orbit, they are falling freely and so their scale, or apparent, weight is zero.
Something professors extol and students shun. Mental work, that is. When it comes to defining mechanical work we need physics, although, to be honest, it was probably the engineers of the nineteenth century that did the most to define work due to their efforts to understand machinery.
If you push a 50 lb box 20 ft across a warehouse floor, you no doubt feel you have done some work. If you push a 100 lb box the same distance, you would be justified in believing you did twice the work as before. If you pushed the 50 lb box a distance of 40 ft, you still believe you did twice the work as before. This perception is codified in the idea that work is defined as force times distance. As such it has the SI units of newtons times meters or N·m. N·m is given the name "joule", such that 1 N·m = 1 joule = 1 J.
Work can be viewed as a way to transfer energy from one object or system of objects to another. Positive work (force acts in the same direction as the motion) done on a system adds energy to that system. Negative work (when force acts opposite the motion of the system as when a car brakes) removes energy. Often in the process of doing work, one form of energy is converted into another.
This is really more than a theorem. It is also a way of looking at the relationship between force and energy. A net force acting on an object will cause the object to accelerate. If this force is not perpendicular to the motion, it will change the object's speed. (A force acting perpendicular to the motion causes centripetal, or radial, acceleration and will not change the object's speed, only its direction of motion, nor will it do any work on the object.) A net force that changes an object's speed also changes the value of (1/2)mv2 for the object, the object's kinetic energy. This is viewed by physicists as follows.
The action of doing work transforms one form of energy into another.
In the case of the work-energy theorem, work done by a net force (called
net work) is transformed
into the energy of motion. Work can also transform one form of energy into
forms other than kinetic, such as potential energy
and/or heat. However, any "leftover" work
(net work) produces a
change in kinetic energy, and the amount of change equals the amount of
net work done. If the net work is positive,
the kinetic energy of the object upon which the work is done increases. If the
net work is negative, the kinetic energy decreases. The formula for the
work-energy theorem is
Wnet = ΔK = (1/2)mv2 - (1/2)mvo2.
where "m" is the mass of the object on which the work is done, "vo"
is the speed of the object before the work is performed and "v" is the
speed of the object after the work has been performed.