84. Two uniform beams of equal weight but of unequal length, are placed with their lower ends in contact on a smooth horizontal plane, and their upper ends against smooth vertical planes: shew that in the position of equilibrium the two beams are equally inclined to the horizon.

85. A bowl is formed from a hollow sphere of radius a; it is so placed that the radius of the sphere drawn to each point in the rim makes an angle a with the vertical, and the radius drawn to a point A of the bowl makes an angle B with the vertical: if a smooth uniform rod remains at rest when placed with one extremity at A, and with a point in its length on the rim of the bowl, shew that the length of the rod is 4a sin ẞ sec(a-ẞ).

DYNAMICS.

I. Velocity.

1. DYNAMICS treats of force producing or changing the motion of bodies.

Before we consider the influence of force on the motion of bodies we shall make some remarks on motion itself: we confine ourselves to the case of motion in a straight line.

2. The velocity of a point in motion at any instant is the degree of quickness of the motion of the point at that instant.

3. If a point in motion describe equal lengths of path in equal times the velocity is called uniform or constant. Velocity which is not uniform is called variable.

4. Uniform velocity is measured by the length of path described in the unit of time. We may take any unit of time we please; and a second is usually chosen. We may also take any unit of length we please: and a foot is usually chosen. Thus by the velocity 16 we mean the velocity of a point which moves uniformly in such a manner that the length of path described in one second is sixteen feet. The word space is used as an abbreviation of the term length of path: thus in the example just given it would be said that the space described in one second is sixteen feet.

5. If a point moving with the uniform velocity v describe the space s in the time t, then s=vt.

For in one unit of time v units of space are described, and therefore in t units of time vt units of space are described; therefore s=vt.

6. Variable velocity is measured at any instant by the space which would be described in a unit of time, if the velocity were to continue during that unit the same as it is at the instant considered.

Hence, as in Art. 5, if v denote the measure of a variable velocity at any instant, a point moving for the time t with this velocity would describe the space vt.

7. The mode of measuring variable velocity is one with which we are familiar in practice. Thus a railway train may be moving with variable velocity, and yet we may say that at a certain instant it is moving at the rate of 30 miles an hour; we mean that if the train were to continue to move for one hour with just the same speed as at the instant considered it would pass over 30 miles.

8. The illustration just employed suggests that a velocity may be given expressed in any units of time and space; it is easy to express the velocity in terms of the standard units.

For example, suppose that a body is moving at the rate of 30 miles an hour. The body here is moving at the rate of 30 × 5280 feet in an hour, that is, in 60 × 60 seconds: 30 x 5280 hence it is moving at the rate of feet in one 60 × 60 second, that is, at the rate of 44 feet in one second. Hence 44 denotes the velocity when expressed in the standard units.

In like manner we may pass from the standard units to any other units.

For example, if v denote a velocity when a second is taken as the unit of time, the same velocity will be denoted by 600 when a minute is taken as the unit of time. For to

say that a body is moving at the rate of feet per second is equivalent to saying that it is moving at the rate of 60v feet per minute.

In like manner if we wish to take a yard for the unit of length instead of a foot, as well as a minute for the unit of time instead of a second, the velocity denoted by v with 60v the standard units will now be denoted by

3

Generally, let v denote a velocity when a second is the unit of time, and a foot is the unit of length; then if we take m seconds as the unit of time, and n feet as the unit of length, the same velocity will be denoted by n

то

EXAMPLES.

1. Compare the velocities of two points which move uniformly, one through 5 feet in half a second, and the other through 100 yards in a minute.

2. Compare the velocities of two points which move uniformly, one through 720 feet in one minute, and the other through 3 yards in three quarters of a second.

3. Two points move uniformly with such velocities that when they nove in the same direction the distance between them increases at the rate of 5 feet per second; and when they move in opposite directions the distance between them increases at the rate of 25 feet per second: find the velocity of each.

4. A railway train travels over 100 miles in 2 hours; find the average velocity referred to feet and seconds.

5. One point moves uniformly round the circumference of a circle, while another moves uniformly along the diameter: compare their velocities.

T. M.

14

II. The First and Second Laws of Motion.

9. The science of Dynamics rests on certain principles which are called Laws of Motion. Newton presented them in the form of three laws; and we shall follow him.

It is not to be expected that a beginner will obtain a clear and correct idea of these laws on reading them for the first time; but as he proceeds with the subject and observes the applications of the laws he will gradually discover their full import. In like manner a beginner of geometry rarely comprehends at first all that is meant by the definitions, postulates, and axioms; but the imperfect notions with which he starts are corrected and extended as he studies the propositions.

In the present Chapter we shall chiefly discuss the First Law of Motion.

10. First Law of Motion. Every body continues in a state of rest or of uniform motion in a straight line, except in so far as it may be compelled to change that state by force acting on it.

It is necessary to limit the meaning of the word motion in the First Law. By the motion of a body is here meant that kind of motion in which every point of the body describes a straight line; in other words, there is to be no rotation. The rotation of bodies is discussed in works which treat of the highest branches of dynamics, and many important results are demonstrated: for example, it is shewn that if a free sphere of uniform density be rotating about a diameter at any instant, it will continue to rotate about that diameter if no force act on it.

In order to exclude all notion of rotation, some writers use the word particle instead of body in enunciating the First Law of Motion.

We must now proceed to consider the grounds on which we rest our belief in the truth of the First Law of Motion.