30. An inelastic particle falls from a height on to a smooth inclined plane. How does it move after impact?

The particle falls from any point in a perpendicular to the plane through its highest point. Show that the time of arrival at the lowest point of the plane is invariable. (25) 31. A particle describes the perimeter of a regular polygon of n sides with a constant velocity V. Show that, when it comes to an angular point, there must be impressed on it a velocity

directed towards the centre of the circumscribing circle.

Hence show that when a particle (mass m) moves in a circle (radius r) with a constant velocity (v), it must be acted on by a force of

mv2

units,

directed towards the centre.

(30)

32. Two particles, whose masses are P and Q, are connected by a rigid rod, the mass of which is put out of the question, the system is placed on a smooth table, and is made to turn round a fixed point O in the rod. Find the position of O, with respect to P and Q, when there is no resultant force at 0. (20)

Stage 3.

You are not permitted to answer more than eight questions.

41. ABCDE is a frame of five equal bars, kept in the form of a regular pentagon by two bars AC, AD. The frame is hung up by the point A, and carries equal weights (W) at B and E. Find the stresses in the bars, putting out of the question the weight of the bars, and the friction of the joints.

Explain how the results would be altered if the weights were hung at C and D instead of at B and E.

(40)

42. Draw an isosceles triangle ABC, and from C draw CD at right angles to the base BC; also draw a circle to touch AC and CD. Suppose that ABC represents a cross section of a beam lying on the ground, and that the circle represents a cross section of a cylinder resting between the beam and a wall CD. Taking account of the friction between the beam and the ground, but not of the friction between the cylinder and the beam or the wall, find the relation between the weights of the beam and the cylinder, when the beam is just beginning to slide out. (30)

43. Show that any system of forces, acting on a rigid body, can be replaced by a single resulting force acting at any chosen point and a couple.

Give further information concerning this force.

Define Poinsot's central axis, and show how to construct it for any given system of forces.

(40)

44. Investigate the attraction of a thin homogeneous circular plate of radius a at a point which is at a perpendicular distance c from the centre of the plate.

Remark upon the cases:-

(1) a infinite, c finite.

(2) a finite, c infinitesimal.

(30)

45. A spherical shell of uniform density attracts an external particle according to the law of gravity; find the resultant attraction.

A sphere of uniform density attracts a particle, the mass of which is 1 lb., at a distance of 4,000 miles from its centre, with a force of 32 poundals. Find the force with which it would attract an equal particle (P) placed at a distance of 60 × 4,000 miles.

=

If P describes a circle round the centre of the sphere, find the periodic time. (N.B., 110 10.4881.) (30) 46. Define the potential of a system of attracting or repelling masses at any point.

Mass, attracting according to the law of nature, is uniformly distributed on the circumference of a circle. Prove that the chord of contact of tangents drawn from an external point divides the mass into two parts having equal potentials at the point. (35)

47. Prove that, at any point of the path of a moving particle, the normal component of the acceleration is

v2R,

where is the velocity and R the curvature at the point. Deduce its expression in terms of x, y and t.

(30)

48. What is meant by the motion of a lamina in its own plane? Show that a lamina can be moved in its own plane from any one position to another by a rotation round a point in that plane. Point out any case that presents an apparent exception.

The motion of a body at any instant can be represented by two angular velocities round parallel axes; find a simpler mode of representing the motion.

49. Define the principal axes of a rigid body.

(30)

If the body be a plane lamina, explain why one of the principal axes at any point must be at right angles to the plane.

If the lamina be an equilateral triangle of uniform density, find the principal axes at one of the angular points. Find also the principal moments of inertia at that point.

(35)

50. Find the period of a complete double oscillation of a compound pendulum when the angular swing is small. How must the pendulum be suspended to make the period a minimum?

(35)

51. Find an expression for the kinetic energy of a body moving, in any given way, in one plane.

AB is an inclined plane and BC is the horizontal plane, through the lowest point B, and both planes are smooth. A uniform rod is placed on AB with one end at B, and is allowed to slide down. Find its angular velocity just before its upper end leaves the inclined plane. (40)

52. A heavy rod is constrained to slide in a vertical line with its lower end on the curved surface of an equally heavy smooth hemisphere, the hemisphere sliding on a smooth horizontal plane. Determine the motion.

Solve the equations of motion for the case in which initially the rod is very nearly in its highest position. (45)

Division II.-FLUIDS.

Stage 1.

You are not permitted to answer more than seven questions. The examination in this subject lasts for three hours.

1. Define the centre of gravity of a body, or system of bodies. State where the centre of gravity is situated in the case of (a) a uniform rod, (b) a cylinder of uniform density, (c) a triangular board.

Show how to find by construction the centre of gravity of three different weights, placed severally at the angular points of a triangle.

(12)

2. Write down the equation of work and energy to meet the case of a body which is subject to the force of gravity and to no other.

If the body, 10 lbs. in weight, falls freely from rest, how much work is done during the time that its velocity changes from 30 to 34 feet per second?

(12)

3. What is the unit of pressure in the foot-pound system of units? Express a pressure of a pound weight per square inch in terms of such unit.

(10)

4. How is pressure measured at any point of a fluid ?

There is a hole in the bottom of a ship 15 feet below the water line, and a force of 168 lbs. has to be applied to a piece of wood held against the hole in order to keep the water out; assuming that a cubic foot of sea-water weighs 64 lbs., find the size of the hole.

5. State the conditions of equilibrium of a floating body.

(12)

The specific gravities of lead and cork may be taken as 11.5 and 0.25. A piece of cork, volume 14 cubic inches, has a piece of lead tied to it by a fine thread; the composite body will just sink in water; find the volume of the lead.

If the composite body were floated in a liquid of specific gravity 1.5, find how much of the cork would be above the surface.

(14)

6. What is meant by the "resultant vertical thrust" on a body wholly immersed in fluid?

Show how its magnitude and line of action may be found. Take as an example a hemisphere of one foot radius just immersed in water with its base vertical. (14)

7. A deal rod of uniform cross section has one end fastened by a thread to the bottom of a vessel, which is gradually filled with water. At first the rod will float, but when the depth of the water exceeds (by not too much) the length of the string, the rod will take an inclined position and the thread will be vertical. Explain why this should be so.

(16)

8. Explain how to find the specific gravity of a body by means of a balance; also explain how the method has to be modified when the body, left to itself, floats in water. In the latter case find a formula for the specific gravity.

(12)

9. What is meant by the Boiling Point of a thermometer? In what sense can it be regarded as a fixed point?

How is the distance between the freezing and boiling points divided in the thermometers in common use?

When a Fahrenheit's thermometer reads 23.9°, what is the reading of the Centigrade thermometer?

(12)

10. State Boyle's Law, and describe briefly its experimental verification.

Two globes are connected by a short pipe furnished with a stop-cock. One of them (A) contains air under a pressure of 30 in. of mercury; in the other (B) there is a vacuum. The diameters of A and B are 8 in. and 10 in. respectively. If the stop-cock is opened, what will be the change in the pressure of the air?

(14)

11. Given that 100 cubic inches of air weigh 31 grains when the barometer is at 30 inches and the Fahrenheit thermometer at 60°, determine the volume of 1 oz. of air when the barometer is at 29.5 inches and the thermometer at 84°.

12. Describe some form of hydrometer.

(14)

In a cylinder, three-fourths filled with water, a hydrometer is observed to rest at a certain depth; suppose the vessel filled up with a fluid of specific gravity 3 and the fluids mix, find what weight must be added to the hydrometer to make it sink to the same depth as before. (16)

Stage 2 and Stage 3 in Fluids, and Honours in Solids and Fluids.

You may take Stage 2, or Stage 3, or, if eligible, Honours, but you must confine yourself to one of them.

The examination in this subject lasts for three hours.

Stage 2.

You are not permitted to answer more than eight questions.

21. ABC is an isosceles triangle, having a right angle at A; forces of 5, 4, and 10 units act respectively from A to B, B to C, and A to C. Find their resultant by means of a diagram drawn to scale. Verify the result by showing from your diagram that the resultant and the force of 4 units have equal moments about A.

(20)

22. State briefly the meaning of the words mass, velocity, acceleration, and define an absolute unit of force.

A mass of 12 lbs. moves in a straight line acted on by a constant force; it acquires a velocity of 50 feet a second in five seconds; how many British absolute units (or poundals) are there in the force? How many footpoundals of work were done in those five seconds? 23. Explain the utility of the notion of "Moment of Inertia." Find the moment of inertia of a hemisphere about a diameter of its base.

24. What are "buoyancy

and "

reserve of buoyancy"?

(20)

(25)

A homogeneous body, of specific gravity s, floats in water with a certain water line. If the body be inverted and its specific gravity changed to 1-s, show that it will float with the same water line as before.

(20)