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An Elementary Treatise on Theoretical Mechanics
By J. H. Jeans Dover Publications, Inc.
Copyright © 2005 Dover Publications, Inc.
All rights reserved.
ISBN: 978-0-486-17469-3
CHAPTER 1
REST AND MOTION
INTRODUCTION
1. Uniformity of nature. If we place a stone in water, it will sink to the bottom; if we place a cork in water, it will rise to the top. These two statements will be admitted to be true not only of stones and corks which have been seen to sink or rise in water but of all stones and corks. Given a piece of stone which has never been placed in water, we feel confident that if we place it in water it will sink. What justification have we for supposing that this new and untried piece of stone will sink in water? We know that millions of pieces of stone have at different times been placed in water; we know that not a single one of these has ever been known to do anything but sink. From this we infer that nature treats all pieces of stone alike when they are placed in water, and so feel confident that a new and untried piece of stone will be treated by the forces of nature in the same way as the innumerable pieces of stone of which the behavior has been tested, and hence that it will sink in water. This principle is known as that of the uniformity of nature; what the forces of nature have been found to do once, they will, under similar conditions, do again.
2. Laws of nature. The principle just stated amounts to saying that the action of the forces of nature is governed by certain laws; these we speak of as laws of nature. For instance, if it has been found that every stone which has ever been placed in water has sunk to the bottom, then, as has already been said, the principle of uniformity of nature leads us to suppose that every stone which at any future time is placed in water will sink to the bottom; and we can then announce, as a law of nature, that any stone, placed in water, will sink to the bottom.
That part of science which deals with the laws of nature is called natural science. Natural science is divided into two parts, experimental and theoretical. Experimental science tries to discover laws of nature by observing the action of the forces of nature time after time. Theoretical science takes as its material the laws of nature discovered by experimental science, and aims at reducing them, if possible, to simpler forms, and then discovering how to predict from these laws what the action of the forces of nature will be in cases which have not actually been subjected to the test of experiment. For example, experimental science discovers that a stone sinks, that a cork floats, and a number of similar laws. From these theoretical physics arrives at the simple laws of nature which govern all phenomena of sinking or floating, and, going further, shows how these laws enable us to predict, before the experiment has been actually tried, whether a given body will sink or float. For instance, experimental science cannot discover whether a 50,000-ton ship will float or sink, because no 50,000-ton ship exists with which to experiment. The naval architect, relying on the uniformity of nature, on the laws of nature determined by experimental science, and on the method of handling these laws taught by theoretical science, may build a 50,000-ton ship with every confidence that it will behave in the way predicted by theoretical science.
3. The science of mechanics. The branch of science known as mechanics deals with the motion of bodies in space, and with the forces of nature which cause or tend to cause this motion. The laws of nature which govern the action of these forces and the motion of bodies have long been known, and were reduced to their simplest form by Newton. Thus we may say that experimental mechanics is a completed branch of science.
The present book deals with theoretical mechanics. We start from the laws supplied by experimental mechanics, and have to discuss how these laws can be used to predict the motion of bodies, — for instance, the falling of bodies to the ground, the firing of projectiles, the motion of the earth and the planets round the sun. An important class of problems which we shall have to discuss will be those in which no motion takes place, the forces of nature which tend to cause motion being so evenly balanced that no motion occurs. Such problems are known as statical.
MOTION OF A POINT
4. State of rest. Before we can reason about the motion of a body we have to determine what is meant by a body being at rest. In ordinary language we say that a train is at rest when the cars are not moving over the rails. We know, however, that the train, in common with the rest of the earth, is not actually at rest, but moving round the sun with a great velocity. Again, a fly crawling on the wall of a railway car might in one sense be said to be at rest, if it remained standing on the same spot of the wall. The fly, however, would not actually be at rest; it would share in the motion of the train over the country, the country would share in the motion of the earth round the sun, and the sun would share in the motion of the whole solar system through space.
These instances will show the necessity of attaching a clear and exact meaning to the conceptions of rest and motion. Obviously our statements would have been exact enough if we had said that in the first case the train was at rest relatively to the earth, and that in the second case the fly was at rest relatively to the car.
5. Frame of reference. Thus we find it necessary, before discussing rest and motion, to introduce the conception of a frame of reference. The earth supplied a frame of reference for the motion of the train, and when a train is not moving over the rails we may say that it is at rest, the earth being taken as frame of reference. So also we could say that the fly was at rest, the car being taken as frame of reference. Obviously any framework, real or imaginary, or any material body, may be taken as a frame of reference, provided that it is rigid, i.e. that it is not itself changing its shape or size.
We may accordingly say that a point is at rest relatively to any frame of reference when the distance of the point from each point of the frame of reference remains unaltered.
6. Motion relative to frame of reference. Having specified a frame of reference, we can discuss not only rest but also motion relative to the frame of reference. When the train has moved a mile over the tracks we say that it has moved a mile relatively to its frame of reference, the earth. When the fly has crawled from floor to ceiling of the car we say that it has moved, say, eight feet relatively to its frame of reference, the car.
In fixing the distance traveled by the fly relatively to the train in an interval between two instants t1, t2, we notice that the actual point from which the fly started is, say, a mile behind the present position of the train; but the point from which we measure is the point which occupies the same position in the car at time t2 as this point did at time t1. So, in general, to fix the distance moved relatively to a given frame of reference in the interval between times t1 and t2, we first find the point A which stands in the same position relative to the frame of reference at time t2 as did the point from which the moving point started at time t1. The distance from this point A to the point B, which is occupied by the moving point at instant t2, is the distance moved relatively to the moving frame of reference.
By the motion of a particle B relative to a particle A, is meant the motion of B relative to a frame of reference moving with A.
7. Composition of motions. Suppose that in a given time the moving point moves a certain distance relatively to its frame of reference, while this frame of reference itself moves some other distance relatively to a second frame of reference, — as will, for instance, occur if a fly climbs up the side of a car while the car moves relatively to the earth.
Let us suppose that there is a frame of reference moving in the plane of the paper on which fig. 1 is drawn, and that the paper itself supplies a second frame of reference. Suppose that the moving point starts at A, and that during the motion that point of the first frame of reference which originally coincided with the moving point has moved from A to B, while the point itself has moved to C. Then the line AB represents the motion of frame 1 relative to frame 2, while BC represents the motion of the moving point relative to frame 1. The whole motion of the point relative to frame 2 is represented by AC. The motion AC is said to be compounded of the two motions AB, BC, or is said to be the resultant of the two motions. Thus:
[ILLUSTRATION OMITTED]
If a point moves a distance BC relatively to frame 1, while frame 1 moves a distance AB relatively to frame 2, the resultant motion of the point relative to frame 2 will be the distance AC, obtained by taking the two distances AB, BC and placing them in position in such a way that the point B at which the one ends is also the point at which the other begins.
There is a second way of compounding two motions. Let x, y represent the two motions. The rule already obtained directs us to construct a triangle ABC, to have x, y for the sides AB, BC, and then AC will be the motion required. Having constructed such a triangle ABC, let us complete the parallelogram ABCD by drawing AD, CD parallel to the side of the triangle. Then AD, being equal to BC, will also represent the motion y, so that we may say that the two edges of the parallelogram which meet in A represent the two motions to be compounded, while the diagonal AC through A has already been seen to represent the resultant motion. Thus we have the following rule for compounding two motions x, y:
[ILLUSTRATION OMITTED]
Construct a parallelogram ABCD such that the two sides AB, AD which meet in A represent the two motions x, y to be compounded, as regards both magnitude and direction; then the diagonal AC which passes through A will represent the resultant obtained by compounding these two motions.
VELOCITY
8. Uniform and variable velocity. Velocity means simply rate of motion. It may be either uniform or variable. If a point moves in such a way that a feet are described in each second of its motion, no matter which second we select, we say that the velocity of the point is a uniform velocity of a feet per second. If, however, the point moves a feet in one second, b feet in another, c feet in a third, and so on, we cannot say that any one of the quantities a, b, or c measures the velocity. The velocity is now said to be variable: it is different at different stages of the motion. To define the velocity at any instant, we take an infinitesimal interval of time dt and measure the distance ds described in this time. We then define the ratio ds/dt to be the velocity at the instant at which the interval dt is taken. If the velocity is uniform, ds/dt is the space described in unit time, and so the present definition of velocity becomes the same as that already given.
Average velocity. If a point moves with variable velocity, and describes a distance of a feet in t seconds, we speak of a/t as the "average velocity" of the moving point during the time t. This average velocity is the velocity which would have to be possessed by an imaginary point moving with uniform velocity, if it were to cover the same distance in time t as the actual point moving with variable velocity.
Units. In measuring a velocity we need to speak in terms of a unit of length and of a unit of time; for instance, in saying that a point has a velocity of a feet per second we have selected the foot as unit of length and the second as unit of time. We can find the amount of this same velocity in other units by a simple proportion.
Thus suppose it is required to express a velocity of a feet per second in terms of miles and hours.
The point moves a feet in one second, and therefore a × 60 × 60 feet in one hour, and therefore
[a × 60 × 60]/[3 × 1760] = 15a/22 miles
in one hour. Thus the velocity is one of 15/a/22 miles per hour.
EXAMPLES
1. A railway train travels a distance of 918 miles in 18 hours. What is its average velocity in feet per second?
2. Compare the velocities of a train and an automobile which move uniformly, the former covering 100 feet a second and the latter 1500 yards a minute.
3. A man runs 100 yards in 9 1/5 seconds. What is his average speed in miles per hour?
4. The two hands of a town clock are 10 and 7 feet long. Find the velocities of their extremities.
5. Taking the diameter of the earth as 7927 miles, what is the velocity in foot-second units of a man standing at the equator (in consequence of the daily revolution of the earth about its axis)?
6. Two trains 230 and 440 feet long respectively pass each other on parallel tracks, the former moving with twice the speed of the latter. A passenger in the shorter train observes that it takes the longer train three seconds to pass him. Find the velocities of both trains.
9. Composition of velocities. All motion, as we have seen, must be measured relatively to a frame of reference. Thus velocity, or rate of motion, must also be measured relatively to a frame of reference. A point may have a certain velocity relative to a frame of reference, while the frame of reference itself has another velocity relative to a second frame. It may be necessary to find the velocity of the moving point with reference to the second frame, in other words, to compound the two velocities.
To do this we consider the motions which take place during an infinitesimal interval of time dt. Let the moving point have a velocity v1 in a direction AB relative to the first frame, while the frame has a velocity v2 in a direction AC relative to the second frame. Then in time dt the moving point describes a distance v1dt, say the distance AD, along AB relative to the first frame, while the frame itself describes a distance v2dt, say AE, along AC relative to the second frame. Let AF be the diagonal of the parallelogram of which AD, AE are two edges; then AF will be the resultant motion of the point in time dt relative to the second frame. Since the moving point describes a distance AF in time dt, the resultant velocity will be AF/dt.
Let us now agree that velocities are to be represented by straight lines, the direction of the line being parallel to that of the velocity and its length being proportional to the amount of the velocity, the lengths being drawn according to any scale we please; for example, we might agree that every inch of length is to represent a velocity of one foot per second, in which case a velocity of three feet a second will be represented by a line three inches long drawn parallel to the direction of motion.
In fig. 3 let Ap, Aq represent the velocities v2, v1 drawn on any scale we please. Since the scale is the same for both, we have
Ap: Aq = v2 : v1.
Now AE = v2dt, AD = v1dt, so that
AE: AD = v2 : v1,
and hence
Ap: Aq = AE: AD.
If we complete the parallelogram Aprq, the diagonal Ar will pass through F, and we shall have
Ar: Ap = AF: AE.
If V is the resultant velocity, it has already been seen that
V = AF/dt,
so that
AF: AE = Vdt:v2dt = V: v2,
and hence
Ar: Ap = V: v2.
Thus Ar represents the magnitude of the velocity V on the same scale as that on which Ap represents the velocity v2. Also since Ar is in the direction of AF, the resultant motion, we see that Ar represents the velocity V both in magnitude and direction. We have accordingly proved the following theorem:
THEOREM. If two velocities are represented in magnitude and direction by the two sides of a parallelogram which start from any point A, then their resultant is represented in magnitude and direction on the same scale by the diagonal of the parallelogram which starts from A.
This theorem is known as the parallelogram of velocities. We may illustrate its meaning by two simple examples.
1. Suppose that a carriage is moving on a level road with velocity V. As a first frame of reference let us take the body of the carriage; as a second frame take the road itself. The velocity of frame 1 relative to frame 2 is then V. Relatively to frame 1, the center of any wheel P is fixed, so that any point on the rim describes a circle about P. Relatively to frame 1 the road is moving backward with velocity V, so that if there is to be no slipping between the rim and the road, the velocity of any point on the rim, relative to the first frame (the carriage), must be V. Thus the velocity of any point Q on the rim relative to frame 1 will be a velocity V along the tangent QT. Representing this by the line QT, the velocity of the carriage relative to the road is represented by an equal line QH parallel to the road. Thus the resultant velocity of the point Q is represented by the diagonal QS of the parallelogram QHST. Clearly its direction bisects the angle HQT. Let L be the lowest point of the wheel, and let X complete the parallelogram QPLX. Obviously this parallelogram is similar to the parallelogramQTSH, corresponding lines in the two parallelograms being at right angles. Thus
QS: QT = QL: QP.
(Continues...)
Excerpted from An Elementary Treatise on Theoretical Mechanics by J. H. Jeans. Copyright © 2005 Dover Publications, Inc.. Excerpted by permission of Dover Publications, Inc..
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