DYNAMICS OF A PARTICLE.

CHAPTER I.

KINEMATICS.

1. Dynamics is the Science which investigates the action of Force; and naturally divides itself into two parts as follows.

2. Force is recognized as acting in two ways: in Statics so as to compel rest or to prevent change of motion, and in Kinetics so as to produce or to change motion.

3. In Kinetics it is not mere motion which is investigated, but the relation of forces to motion. The circumstances of mere motion, considered without reference to the bodies moved, or to the forces producing the motion, or to the forces called into action by the motion, constitute the subject of a branch of Pure Mathematics, which is called Kinematics. To this, as a necessary introduction, we devote the present chapter.

4. The rate of motion (or the rate of change of position) of a point is called its Velocity. It is greater or less as the space passed over in a given time is greater or less: and it may be uniform, i. e. the same at every instant; or it may be variable.

Uniform velocity is measured by the space passed over in unit of time, and is, in general, expressed in feet per second; if very great, as in the case of light, it may be measured in miles per second. It is to be observed, that Time is here used in the abstract sense of a uniformly-increasing quantity T. D. 46

1

-what in the differential calculus is called an independent variable. Its physical definition is given in Chap. II.

5. Thus, a point moving uniformly with the velocity v describes a space of v feet each second, and therefore vt feet in t seconds, t being any number whatever. Putting s for space described in t seconds, we have

the

s = vt.

Hence with unit velocity a point describes unit of space in unit of time.

6. It is well to observe that since, by our formula, we have generally

and since nothing has been said as to the magnitudes of s and t, we may take these as small as we choose. Thus we get the same result whether we derive v from the space described in a million seconds, or from that described in a millionth of a second. This idea is very useful, as it will give confidence in results when a variable velocity has to be measured, and we find ourselves obliged to approximate to its value by considering the space described in an interval so short, that during its lapse the velocity does not sensibly alter in value.

7. Velocity is said to be variable when the moving point does not describe equal spaces in equal times. The velocity at any instant is then measured by the space which would have been described in a unit of time, if the point had moved on uniformly for that interval with the velocity which it had at the instant contemplated.

Letv be the velocity of the point at the time t, measured from a fixed epoch, s the space described by it during that time, and s+ds the space described during a greater interval t+St. Suppose v, to be the greatest, and v, the least, velocity with which the point moves during the time St; then vst, vst would be the spaces which a point would describe in that interval, moving uniformly with these velocities re

spectively. But the actual velocity of the point is not greater than y1, and not less than v,, therefore as regards the actual space described,

ds is not greater than vôt, and not less than vôt,

however small St may be. But, as St continually diminishes, V1 and V2 tend continually to, and ultimately become each equal to, v. Therefore, proceeding to the limit,

If v be negative in this expression, it indicates that s diminishes as t increases; the positive case, which we have taken as the standard one, referring to that in which s and t increase together. It follows that, if a velocity in one direction be considered positive, in the opposite direction it must be considered negative; and consequently the sign of the velocity indicates the direction of motion.

8. It will be easily seen that the idea of velocity explained above is equally applicable whether the point be considered as moving in a straight, or in a curved, line. In the latter case, however, the direction of motion continually changes; and it will be necessary to know at every instant the direction, as well as the magnitude, of the point's velocity. This is done by considering the velocities of the point parallel to the three co-ordinate axes respectively. For, if the co-ordinates of the point be represented by x, y, z, the rates of increase of these, or the velocities parallel to the corresponding axes, will by reasoning analogous to the above be dx dy dz dt' dt' dt'

Denoting by v the whole velocity of the point, we have

and, if a, B, y be the angles which the direction of motion

makes with the axes,

to the axes, and are therefore called the Component Velocities of the point and, with reference to them, v is called the Resultant Velocity.

9. It follows from the above, that, if a point be moving in any direction, we may suppose its velocity to be the resultant of three coexistent velocities in any three directions at right angles to each other; or, more generally, in any three directions not coplanar. But the rectangular resolution is the simplest and best except in some very special questions.

Let vx, vy, v2 be the rectangular components of the velocity v of a moving point, then the resolved part of v along a line inclined at angles λ, μ, v to the axes will be

vx cos λ + v2 cos μ + v2 cos v.

For, let a, B, y be the angles which the direction of the point's motion makes with the axes, the angle between this direction and the given line. Then the resolved part of v along that line is

v cose

' = v {cos a cos λ + cos ẞ cos μ + cos y cos v}

= vx cos λ + v12 cos μ + v2 cos v.

10. These propositions are virtually equivalent to the following obvious geometrical construction: