where x, y, z may have any positive values subject to the condition

1. Solve the problem of the oblique impact of two elastic spheres of coefficient e.

2. A smooth ball is bounced near the edge of one of a flight of steps of height h and coefficient of restitution being e.

breadth b the Shew that if

the horizontal and vertical velocities just before the first impact are b√g (1 − e)/2h (1 + e) and √2gh /(1 − e2) the ball will go down the flight making each step in exactly the same way.

3. Define simple harmonic motion, and investigate the force required to produce it in a particle.

4. Two equal particles of mass m are attached to the middle and one end of a light elastic string of length 21 and modulus A. The string is now hung up by the other end. Shew that the particles can execute simple harmonic oscillations together of the same period and phase, the period (7) being given by

λ2 T4-12T2λlm T2 + 16π1m22 = 0 and the ratio of the amplitudes being

5. A simple pendulum is attached to one end of a horizontal rod which turns with uniform angular velocity about a vertical axis through the other end. The air-resistance to the bob of the pendulum is k times its kinetic energy. If the inclination of the string to the vertical is 45°, shew that the distancer of the bob from the axis is given by the equation k2p4 + 4r2 — 4g2/w1= 0.

6. Prove that the sum of the moments of two forces at a point about a line is equal to the moment of their resultant, and that all the lines through a point about which the moments of a force are the same lie on a right circular cone.

1

7. A regular tetrahedron is formed by three uniform heavy bars smoothly jointed at one end of each, and strings tying the other ends which rest on a smooth horizontal plane.

If the weight of each bar is w, and a weight Wis supported at the vertex, shew that the tension of each string is (2 W + 3w)/6√6.

8. Find the centre of gravity of a semicircular disc.

9. Two equal rough rectangular cards prop each
other up on a rough horizontal plane with the
top of one card just projecting over the top of
the other. Assuming
Assuming a common coefficient of
friction, shew that the cards are on the point of
slipping when their inclination to the vertical is
the angle of friction.

10. Find the centre of pressure of a triangle ABC immersed in heavy liquid with its base BC horizontal, and at a depth h below the surface, the inclination of the triangle to the vertical being a.

11. Find the pressure in heavy liquid rotating as a rigid body around a vertical axis. Treat similarly the case where the axis of rotation is horizontal, and the liquid completely fills a closed vessel.

MIXED MATHEMATICS.-PART II.

FIRST PAPER.

The Board of Examiners.

1. Find the position and magnitude of the resultant of a system of forces in one plane.

2. A plane system of forces is applied at fixed points of a body, the forces being given in direction and magnitude. Shew that if the forces satisfy two conditions the body can be put into a position of equilibrium by turning it around, and find the rotation required.

3. A funicular polygon consists of a light string with weights attached along its length, the points of suspension being on the same level. If T is the tension of a straight piece of the string of length 1, T, the horizontal tension, s the span, W one of the weights, and d its distance below the points of suspension, then

ΣΤΙ = Τ8 + ΣWa

4. Investigate the reduction of a system of forces on a rigid body to a Poinsot's wrench, and find the equations of its axis.

5. Assuming the equation of energy, shew that the condition of static stability, when gravity is the only active force, is that the height of the centre of gravity of the system should be a minimum.

6. A rough sphere of radius a and weight w rests on a fixed sphere of radius b. Part of the weight of w is taken by a string attached to its highest point, which passes over a pulley at a distance h vertically above the point of attachment, and carries a weight W. Shew that the equilibrium

is stable if

wah

W > 2(2a + h) (b + a) — bh'

7. Prove that a light string takes the form of a parabola if the weight suspended from any portion of it is proportional to its horizontal projection.

A uniform heavy string is subject to an outward normal force proportional to the depth below a given level. Prove that if is the inclination to the horizontal, and y the vertical ordinate from a properly chosen line

cos A+ By + Cly,

where A, B, C are constants.

8. Investigate the expressions for the acceleration of a particle in plane polar coordinates.

A smooth light rod turning freely round a fixed peg at one end carries two rings of equal mass connected by a string of length 3a, which passes around the peg. The rod is started off with an angular velocity o with the outer ring at a distance 2a from the peg, its initial velocity along the rod being zero. Shew that if at time t the distance of this ring from the peg is r then