8. For the illustration of these principles consider the motion of a circular hoop C, rolling on another cylinder O, which is fixed. The rolling of the hoop from position (1) to

(1)

0

an infinitely near position (2) may be considered as taking place in one of three principal ways.

The whole hoop may rotate through a small angle round O, thus coming into position (3), and then a rotation round C will bring the hoop into position (2).

(3)

Secondly:

The hoop may be translated into position (4), and then brought by a rotation round C to position (2).

By one rotation round the instantaneous centre. This point must lie in OC; for OC is at right angles to the direction of motion of C. To get another line on which it lies, consider the motions of points very near to N. These are moving away from or towards O. Hence N must be the instantaneous centre. That the body is at the moment rotating round N will be better seen by looking on the circles as many-sided polygons of equal sides. Each angular point becomes in turn the centre of rotation. But the axis is continually changing, and if the question considered be one of change of velocity, the motion must not be considered as if it were round a fixed point at N.

If be the angular velocity of revolution of C about O, measured by the angle at O, the velocity of rotation of the hoop round C in the first method will be measured by the angle at C; and since the arcs of the two circles which have been in contact are equal, it will be

ON
Ω
CN'

If we combine these by Art. 6, they give a resultant rotation round N.

In the second method the linear velocity is measured by the distance the centre has moved, i.e. by OC., and the angular velocity by the angle between CN (fig. 4) and CN (fig. 2), i.e. by (< 0 + ‹ C) (fig. 2).

To reduce this to either of the others, consider the linear velocity OC. as the resultant of two equal and opposite angular velocities.

The single angular velocity in the third method is the same as in the second, but it is about N. For the whole rotation must be measured by the same angle whatever be the axis. Or we may see it thus. Calling it w, the linear velocity of C due to it is w CN, and this must be the same as that found by the last method, viz. Q.OC.

EXAMPLES.

1. If two points are rigidly connected, their velocities in the direction of the straight line joining them are equal.

2. A mirror rotates about a vertical axis with an angular velocity w, and a ray of light falls on it from a distant fixed point on the horizon. What is the angular velocity of the reflected ray?

3. Express a velocity of 100 revolutions a minute in units of angular velocity.

4. Compare the velocity of rotation of the earth with the mean angular velocity of revolution of its centre.

5. If v be the linear velocity in Art. 5, which is equivalent to the angular velocities w, w about A and B, shew that v = AB. w.

6. Where is the instantaneous centre when a ladder is slipping down in a vertical plane between a wall and the ground?

7. The paddle-wheel of a steamboat is rotating with velocity w, and the vessel is moving with velocity v; where is the instantaneous axis of the paddle-wheel?

8. Prove that any motion of a rigid body of which the points move in parallel planes may be represented by supposing a right cylinder fixed in the body to roll on a right cylinder fixed in space.

9. What are these cylinders in the case of question 7?

10. If a straight rod be moving in any manner in a plane, the directions of motion of all its points will in general touch a parabola.

11. AP, BQ are two arms moveable round the fixed centres A, B; and the points P and Q are connected by a link (rod) PQ; shew that the angular velocities of the arms AP, BQ are inversely proportional to the segments into which the link, or its direction produced, divides AB.

II.

GEOMETRY OF MOTION.

1. We now come to the case of simultaneous rotations about axes inclined to one another. The motions of points are no longer in one plane or in parallel planes. It will be necessary to represent the axes themselves. The way in which. an angular velocity is geometrically represented is as follows: take the axis xx'; on it take a point 0; let Ox be the direction which is considered positive. Place a watch at O with its face towards x. A rotation whose direction coincides with the direction of motion of its hands is considered positive. It is measured by the line OA, which contains as many units

of length as the angular velocity contains units of angular velocity. An angular velocity in the opposite direction is represented by a straight line measured along Ox'.

With this convention a positive angular velocity round Ox-one of a rectangular system of axes as usually drawnwill tend from Oy to Oz; one round Oy from Oz to Ox; one round Oz from Ox to Oy.

The basis of this subject is the proposition called the parallelogram of angular velocities, which is:

If a body have simultaneous angular velocities about two inclined intersecting axes, and if these be represented by the adjacent sides of a parallelogram, then shall the resultant angular velocity be about the diagonal which passes through their intersection and proportional to it in magnitude.