MECHANICS I. GENERAL PRINCIPLES 15. Definitions.-Mechanics treats of the motions of masses and of the effect of forces in causing or modifying those motions. It includes those cases where forces cause relative motions of the different parts of an elastic body causing it to change its shape or size, as when a gas is compressed or a spring bent. Such changes in size or shape of different portions of a body are called strains. Bodies which do not suffer strain when acted on by forces are said to be rigid. All known bodies yield more or less to distorting or compressing forces, but when considering the motion of a body as a whole, all bodies in which the strains are small may be regarded as practically rigid. Thus we may treat the motion of a grindstone or of a shell from a rifled gun as though these bodies were rigid, though we know that they are slightly strained by the forces acting. Mechanics is usually subdivided into kinematics and dy namics. Kinematics treats of the characteristics of different kinds of motion, and of the modes of strain in elastic bodies without reference to the forces involved. Dynamics treats of the effect of forces in causing or modifying the motions of masses and in producing strains in elastic bodies. It is usual to treat dynamics under the heads statics and kinetics. Statics is that part of dynamics which deals with bodies in equilibrium or when the several forces that may be involved are so related as to balance or neutralize each other, so far as giving motion to the body as a whole is concerned. Kinetics is that part of dynamics which treats of the effect of forces in changing the motions of bodies. IDEAS AND DEFINITIONS OF KINEMATICS. 16. Motion Relative. it is said to be in motion. When a body is changing its position of a body except by its distance from surrounding objects. When it is said, therefore, that a body has moved, it is always meant that there has been a change in its position with reference to some other objects regarded as fixed, or in other words there has been relative motion. Thus we know only relative motion, and when we speak of an object as at rest we usually mean with reference to that part of the earth's surface in our vicinity. 17. Displacement.-The distance in a straight line from one position of the body to another is called its displacement from the first position. To completely describe any displacement, its amount and direction must both be given. If an extended rigid body is displaced, as when a book is moved on a table, it may be moved in such a way that its edges will remain parallel to their original directions, in which case the displacements of all points in the body will be the same both in amount and direction. The motion is said to be one of simple translation without rotation. But in general when a rigid body is moved there is rotation as well as translation, so that to bring it into the second position from the first we may first imagine it to be translated till some point in the object is brought into its second position. Then by a rotation about a suitable axis through that point the whole body may be brought into the second position. 18. Vectors and Their Representation. All quantities which involve the idea of direction as well as amount are said to be vector quantities or vectors. Such are displacements, velocities, forces, etc. While quantities having magnitude only, without any reference to direction, are known as scalar quantities. Volume, density, mass, and energy are scalar magnitudes. vector quantity is represented by a straight line which indicates by its direction the direction of the vector, and by its length the magnitude of the vector, the length being measured in any convenient units, provided the same scale is used throughout any one diagram or construction. = It must be remembered, however, that a vector represented by a line AB is not the same as that represented by BA, one is the opposite of the other, or AB - BA. This will be evident if AB represent a displacement from A to B. A displacement BA will exactly undo what the other accomplished, and bring the body back to its starting point. The straight line representing a vector is therefore commonly represented with an arrow-head indicating its positive direction. 19. Composition of Displacements.—If a man in a railway car were to go directly across from one side to the other, say from A to B (Fig. 1), then the line AB will represent both in amount B and direction his displacement considered only with respect to the car. But if the car is in motion and in the meantime has advanced through the distance AC, the man will evidently come to D instead FIG. 1. of to B. The displacement of the car with reference to the earth is AC and the displacement of the man relative to the earth is AD. This is called the resultant displacement of the man, of which AB and AC are the components. Another way of stating this is that the man received simul taneously two displacements AB and AC, for if he had not been displaced in the direction AC he would have gone to B, while if he had not had the displacement AB he would have been carried to C. From the above it is evident that the resultant of any number of simultaneous displacements may be found just as if they had been taken successively. For example, let it be required to find the resultant of four displacements repre sented in amount and direction by the vectors A, B, C, D. If A were the only displacement, the body would be brought from 0 to a, but B is also a component displacement, therefore draw B' equal and parallel to B, and the result of the two displacements will be represented by the distance O b. Then in like manner draw C' and D' equal and parallel, respectively, to C and D, and it is clear that the result of the four displacements on a body originally at O would be to transfer it to O'. Therefore, the resultant of the four displacements is the single displacement R, and this is so whether the component displacements occur simultaneously or successively. The particular order in which the several components are taken is quite immaterial. This construction by which the resultant is found is called the diagram of displacements, it is perfectly general and applies whether the components are in the same plane or not. 20. Composition of Vectors.-The above construction is a particular instance of the addition or composition of vectors. By a precisely similar process the resultant of any set of vectors may be obtained whether they represent forces, velocities, momenta, or any other quantities having direction as well as magnitude. 21. Resolution of Displacements.-There is only one resultant displacement that can be found when the components are given, in whatever order they may be taken. If it is required, however, to resolve a given displacement into its components, there are an infinite number of ways in which it may be done. For f example, the displacement AB (Fig. 3) may be regarded as having Ac and CB as its components, or Ad and dB or Ae and eB, or it may be considered the resultant of the three displacements Ag, gh, hB. Or if any broken line whatever be taken starting at A and terminating at B, AB will evidently be the resultant of the displacements which are represented in amount and direction by the several parts of the broken line. FIG. 3. e 22. Resolving of Vectors.-What has been just said of the resolving of a displacement into components is equally true of the resolving of any other vectors whatever into component vectors, and applies to the resolution of velocities, forces, etc. 23. Velocity. The velocity of a body is the rate at which it passes over distance in time. It is a vector quantity, its direction being as important as its amount. The term speed is familiarly used to express the amount of velocity without reference to its direction. Two bodies may be moving with the same speed, but if they are not going in the same direction their velocities are different. This is the strict use of the word velocity, it is often somewhat loosely used to express merely the speed of motion. 24. Constant Velocity.-When a body moves in a straight line always passing over equal distances in equal times it is said to have constant or uniform velocity. It is evident that the motion must be in a straight line, otherwise the direction of the velocity would not be constant. In this case of motion if the length of any part of the path be divided by the time taken for the body to traverse that portion, the result is what is called the rate of motion, or the distance passed over per unit time, and is the same whatever part of the path may be chosen. It is this quantity which is the speed or the amount of the velocity. Thus when a train is moving with constant velocity, the number of miles run in a given time divided by that time expressed in hours, is the speed in miles per hour. 25. Variable Velocity.-When either the rate or direction of motion of a particle is changing, it is said to be moving with variable velocity. Thus the velocity is varying in case of a falling body which constantly gains in speed or in case of a a FIG. 4. railway train rounding a curve where the direction of motion is changing. To understand what is meant by the speed of motion at a particular point when the velocity is constantly changing we may consider a short portion b c (Fig. 4) of the path of the body having at its middle the point a at which the speed is to be determined. Divide the length of b c by the time taken by the body in traversing it. The result will be what may be called the average speed over that part of the path. If, now, the part chosen is taken smaller and smaller, always having the given |