point O does not move at all; then a point B, nearer to O than C is, moves slower than C; while one D outside moves faster.

We require then some different method, and we proceed thus:

Instead of considering the motion of the point C, let us draw the line OC of indefinite length and consider how that moves. The line starts at OC, and as C moves to C' it turns to OC', turning through the angle COC'.

Fig. 30.

Now it does not matter what line in the body we take, we shall find that they have all turned through the same angle.

Take first another line OD through the centre, then OD turns to OD', while OC turns to OC'. Thus C'OD' is only COD in a new position,

.. <C'OD' = <COD,

add to each <COD', and we have

<C'OC=<D'OD.

Next, take any line whatever, CD represents such a line -C and D being any points in OC, OD.

Then the triangle C'OD' is COD in a new position ; and evidently during the motion each side must turn through the same angle.

We see now then that we can completely define the motion of a turning piece by giving the angle turned through by any line in it.

Angular Velocity. When a point moves it traces out a line, and its velocity is measured by the length of line traced out in a unit of time.

When a line swings round a point, as OC (Fig. 30), it traces out an angle, and so we define its velocity by the angle which it traces out in a unit of time. The

angle traced out per unit time gives us then the angular velocity of the line, and therefore also of the body on which it is drawn.

The relative velocity of a turning pair is then an angular velocity, and is, if uniform, measured by the angle turned through per second. The measurement of

a non-uniform velocity has been fully explained in the case of linear velocity, and need not be repeated (pages 21 and 22).

Units of Angular Velocity.-Angle is measured either in English measurement by Degrees, of which 90 form a right angle; in French measurement by Grades, the right angle being divided into 100 parts; in practical work by revolutions or whole turns.

[In this case the student must remember that the angle does not simply refer to the space between the old and new positions of the swinging line, but to the whole space which has been swept out by the swinging line since it commenced swinging. Thus in Fig. 30 OC might go on revolving for a number of turns and finally arrive at OC', the angle traced out would then be C'OC plus the whole of the turns.]

Or, which is best of all for all purposes, by circular measure, which we will now explain. Referring to Fig. 30, let OC be the swinging line; take C, any point in it, then C moves in a circle as OC swings; let OC swing to OC', tracing out the angle C'OC. Then we measure C'OC by the ratio which the arc CC' bears to the radius Oc,

.. Circular measure of C'OC-CC' (arc)

ос

This measure is independent of the position of C. For evidently

and whether we take D or C, or any other point, we get the same numerical value of the circular measure.

It must be clearly understood, however, that in this method of measuring angles we only differ from the other methods in the size of the unit angle. Angles can only be measured in terms of angles, but the unit chosen in this case has certain advantages in simplifying formulæ which makes it superior to the others. Its one disadvantage, if it be one, is that it is large, so that fractions have to be used.

We must now see what this unit angle is.

Unit of Circular Measure.-Being the unit, its value is I,

arc

.. I= radius'

so that the unit subtends at the circumference an arc equal in length to the radius.

This properly defines it, and enables us to compare it with our other units. For example-1° or 1 degree subtends an arc, whose length is 180 degrees in the half circumforence,

π × radius

180

there being,

whence

=

Unit of circular measure=

degrees,

π

=57.3°.

The name Radian has been given to this unit.

Relation between Angular and Linear Velocity. By expressing angular velocity in circular measure we can obtain a simple relation between the angular velocity of a turning piece and the linear velocity of any point in it.

For, referring to Fig. radius OC=r say; let

30, let C be the point at a V = linear velocity of C; A

If now t

= angular velocity of body in circular units. be the number of units of time taken in moving through

Thus we have a simple relation between the two velocities.

One important use of the preceding is to determine the rubbing velocity of a shaft in its bearing. r is then the radius of the bearing, and if r be in feet and A in radians per second, V gives in feet per second the velocity with which the metal of the shaft rubs over that of the bearing.

Radius of Reference. The above also shows us that although the linear velocity of any point C is not sufficient to determine the turning velocity, yet when combined with a statement of the radius at which C is, it is sufficient. The velocities of turning pairs are often stated in this way by giving the linear velocities of points at a certain radius, the radius selected being called the Radius of Reference. Evidently we can in this way compare the velocities of turning pairs, or even of a turning and a sliding pair.

Degrees or grades are never practically used to measure angular velocity, but revolutions per minute is a common unit of measurement, so we must compare this method with the circular unit measurement.

=27 units of circular measure;

.. I revolution per minute=2′′ circular units per minute,

=

..n revolutions per minute=

2π

60

2π12

60

circular units per second;

circular units per second,

or, if A be the circular measure of the same velocity reckoned per second,

A=

2πn

60*

Revolutions per second is not a measure of common

Occurrence.

Screw Motion.-.The two kinds of motion we have just investigated can be represented on a plane; because, although the bodies dealt with have been solid, yet parallel plane sections of them each moved in its own plane, and any one plane section could be taken to fully represent the motion of the whole solid.

The third simple case of motion, viz. Screw Motion, which we are about to consider, consists of motions in perpendicular planes, for while a section perpendicular to the screw axis revolves in its own plane, it also advances along the axis.

The simplest case is that of a common bolt and nut shown in Fig. 31. When we turn the bolt head in the direction of the arrow (b), there ensues, beside the turning motion, a forward motion, i.e. motion to the right,

(b)

in (a).

(a)

Fig. 31.

We may state here that we shall find it convenient to use terms which define the direction of a turning