also since the vertical velocities of P and Q are equal

xii. Find the time in which a particle under the action of gravity describes from rest any arc of a cycloid terminated at the lowest point, the axis of the cycloid being vertical and its vertex downwards.

Hence or from the rider to xi. determine the time of a small oscillation of a simple pendulum.

Pendulums which beat seconds correctly in London (9=32.19) and Edinburgh (g=32-20) respectively are interchanged in station. If started simultaneously from the vertical position towards the left, after how many seconds will they again be both vertical and moving leftwards?

To deduce the time of oscillation of a simple pendulum Ꮎ from the rider to xi., we observe that when a is small, cos expressed by equation (1), differs from unity by a quantity

of the same order as sin2

α

2

(It is easy to obtain the next approximation to the time of oscillation of a simple pendulum.

For if the circumference DQC be divided into n equal parts, n being very large, time over element at Q

Hence when n is odd the series vanishes, and when n is even is reduced to its middle term which is multiplied by an infinitely small quantity.

In either case the time over the whole circumference DQCD

and this is the approximate time of swing of a simple pendulum, length through an angle 2a from rest to rest).

(Second Rider).

The London pendulum, removed to Edinburgh, swings

from rest to rest in

3219
3220,

of a second. The Edinburgh

pendulum, removed to London, swings from rest to rest in

By the condition of the question, the London pendulum, when at Edinburgh, must make two swings more than the other in the same time.

Hence, if t be the time in seconds,

1. DEFINE a fluid and prove that the pressure of a fluid at rest is the same in all directions about a point.

Define the measure of the elasticity of a fluid and prove that if the elasticity is equal to the pressure, the pressure of the fluid is inversely proportional to the volume.

A fluid is a substance such that the smallest shearing stress if continued will cause a constantly increasing change of form.

When the fluid is at rest there must be no shearing stress, and therefore the stress must be uniform in all directions about a point, and this will be true whatever be the degree of viscosity of the fluid. (Maxwell, Heat, p. 276).

The elasticity of a fluid under any given conditions is the ratio of any small increase of pressure to the cubical compression thereby produced.

If v be the volume of a given quantity of gas when the pressure is p, and the volume when the pressure is increased to p', the increase being small, the cubical compression may be measured by

v

v'

Therefore the elasticity is PP, and if this is equal to p,

0- v

This may also be proved by a diagram, as in Maxwell's Heat, pp. 107 and 111.

13-4]

SENATE-HOUSE PROBLEMS AND RIDERS.

27

2. If a solid be immersed in a liquid the resultant pressure on the surface immersed is equal and directly opposed to the weight of the displaced liquid.

Deduce the conditions of equilibrium of a floating body.

If an elliptic lamina with its centre of gravity at an excentric point float in liquid, prove that there may be two or four positions of equilibrium and point out which are stable and which unstable.

The lines of floatation will touch a similar ellipse, and the centres of inertia of the segments cut off by the lines of floatation will also lie on a similar ellipse, the tangent to which at any point will be parallel to the corresponding line of floatation.

If we draw normals from the centre of inertia of the lamina to this last ellipse, these normals will be vertical in the positions of equilibrium.

Two or four normals can be drawn according as the centre of inertia lies outside or inside the evolute of this ellipse.

The points of contact of the normals with the evolute will be the metacentres, and when a normal is vertical the equilibrium will be stable when the point of contact lies above the centre of inertia of the lamina, unstable when it lies below.

3. Find the centre of pressure of a triangular lamina when immersed in liquid (i) with its base in the surface, (ii) with its vertex in the surface and base horizontal.

If a quadrilateral lamina ABCD in which AB is parallel to CD be immersed in liquid with the side AB in the surface, the centre of pressure will be at, the point of intersection of AC and BD if AB2 = 3 CD".

(i) The centre of pressure is at half the depth of the

vertex.

(ii) The centre of pressure is at three-fourths the depth of the base.

Let E, F (fig. 18) be the middle points of AB and CD and G, H the centres of pressure of the ABD, and the ABCD respectively.