3. Two spheres of masses m, m' impinge directly with relative velocity v. Shew that the loss of kinetic

energy is

where e is the coefficient of impact.

An inelastic sphere is projected vertically upwards with velocity v, and a similar sphere is projected with the same velocity in the same straight line when the first sphere is at its highest point. Shew that the two will reach the ground at a time √3v/2g after they meet.

4. Investigate the acceleration of a particle moving uniformly in a circle of radius a with velocity v.

5. Investigate the resultant of two parallel forces on a rigid body.

Three forces on a rigid body act along the sides of a triangle taken in order. Shew that the forces reduce to a couple if, and only if, the forces are proportional to the sides along which they act.

6. State and prove the conditions that a rigid body acted on by three forces only in one plane may be in equilibrium.

A circular wheel stands vertically on a smooth horizontal plane. On the edge of the wheel rest two smooth uniform rods of weights W, W' and lengths 1, ', which are smoothly jointed at one end of each to two points of the plane distant d, d' from the lowest point of the wheel in its plane and on opposite sides.

Shew that for equilibrium

Wld' sin 2a W'l'd sin 2a',

=

where a, α are the inclinations of the rods to the horizon.

7. A heavy uniform rod of length b is kept horizontal with one end pressing against a rough vertical wall by means of a string attached to the other end and to a point of the wall a height h_vertically above the rod. Shew that the coefficient

of friction is at least h/b.

8. Investigate the position of the centre of mass of a solid homogeneous tetrahedron.

9. Assuming the normality of fluid pressure prove its equality in all directions at the same point..

10. Investigate a formula for the total pressure on a plane immersed in a heavy liquid.

Find the total pressure in dynes on a triangle of area 1 sq. cm. whose angles are at depths of 1, 2, and 3 cm. below the surface of water, the barometric height being 76 cm.

11. Prove that the resultant pressure of a heavy liquid on a body immersed in it is equal to the weight of fluid the body displaces.

A cylindrical vessel of weight w, radius r, and height k is placed open end down in water and floats with a length 7 above the surface.

Shew that

g2p2x2+hk = (w + gọπr2l)(w + gpπr2h), where h is the height of the water barometer and the volume of the material of the vessel immersed is neglected.

MIXED MATHEMATICS.-PART II.

The Board of Examiners.

1. A bird which is flying with velocity v in a horizontal straight line at a height his to be hit by a stone thrown with velocity v' when the bird is vertically over the point of projection. Shew that the angle a which the direction of projection makes with the horizon is given by cos a = v/v', and find the position of the bird when struck.

2. Find expressions for the accelerations of a particle along the tangent and normal of a plane curve in which it is moving.

3. Shew that the work done in stretching a string is half the product of the final tension and the extension.

A small smooth heavy ring slides on a circular wire in a vertical plane, and an elastic string of modulus and natural length a, equal to the radius of the circle, is attached to the ring and to the lowest point of the circle. The ring being just displaced from the highest point, find the velocity and the reaction in any position.

4. Shew that the orbit of a particle acted on by a central attractive force varying as the distance from the centre is an ellipse.

Two strings AB, natural lengths a, a B of mass m and to

BC of moduli X, X' and are attached to a particle fixed points A, C, gravity

being neglected. The particle being displaced in the line AC shew that it executes simple harmonic oscillations of period

maa'

2π λa' + X'a'

5. Prove that a particle can describe an elliptic orbit under a force to the focus varying inversely as the square of the distance.

Shew that an increase in the mass of a sun always gives rise to a diminution of the major axis and of the periodic time of a planet.

6. Prove that a system of forces on a rigid body can in general be reduced to a single force and a couple in a plane perpendicular to the force.

Make the reduction in the case of two equal forces Facting along two not-parallel diagonals of opposite faces of a cube of edge a.

7. Find the centre of mass of a homogeneous hemisphere.

If a cylindrical hole of radius r is cut centrally and perpendicular to the base through a hemisphere of radius a, shew, by slicing parallel to the base or otherwise, that the c.m. of the remainder is at a distance a from the

base.

8. Prove the principle of virtual work for a rigid body.

A square formed of four uniform heavy rods smoothly jointed at the corners is suspended from one of the corners and is kept abroad by a light

Y

rod along the horizontal diagonal. Shew that the thrust of this rod is half the weight of the

square.

9. Shew how to transfer the forces which act on the bars of a framework to the joints, examining what alteration is made in the reactions by the transfer.

10. A framework in a vertical plane consists of two equal equilateral triangles DAB, EBC with their bases AB, BC in the same horizontal line, and another horizontal bar from D to E. A, C rest on horizontal supports, and a weight Wis suspended from B. Draw a force diagram to determine the stresses, neglecting the weight of the bars.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. A sphere of weight W and radius a is attached by a light string of length l to a point of a smooth vertical wall against which it rests. A horizontal force Fparallel to the wall is applied to the sphere through the centre. Find the position of equili

brium and the reaction of the wall.

2. Investigate the resultant of two couples not in the same plane.

A system of forces such as (X, Y, Z) at (x,y,z) acts on a body which can turn freely around and slide freely along the line (aa)/l = (y — b)/m =(c)/n. Find the analytical conditions of equilibrium.