UNIVERSITY EXAMINATION PAPERS.

The following are the Papers of Questions actually given in the Examinations for the ordinary degree of B.A. at Cambridge in the years 1860, 1861, 1862, and 1863.

MECHANICS AND HYDROSTATICS.

SATURDAY, May 26, 1860.

FIRST DIVISION—A.

1. Explain clearly what you understand by the Weight of a

body.

A cubic inch of lead is suspended by a spiral spring, and the consequent elongation of the spring observed. If the experiment were repeated at the Equator, would the elongation be the same or different? Give your reasons.

2. If two forces, acting perpendicularly at the extremities of the arms of any Lever, balance each other, they are inversely as the arms.

A grocer buys tea wholesale at the rate of 4s. a-pound, and in weighing it out to his customers uses a Balance, the arms of which are as 16 to 15; at what price must he profess to sell it per pound, in order that he may make a profit of 20 per cent. ?

3. Enunciate the 'parallelogram of forces'; and assuming it to be true, so far as the direction of the Resultant is concerned, complete the proof.

Explain the nature and action of the forces, which are called into play, when a boat is towed down a river by men or horses pulling at a single rope.

4. Investigate the conditions of equilibrium on the Wheel and Axle.

An endless rope, sufficiently rough not to slip, is stretched round two wheels whose radii are as 1 to 3; find the ratio of the radii of the corresponding Axles, in order that equal weights, suspended from strings wound round them in contrary directions, may produce equilibrium.

5. Find the Centre of Gravity of a triangle.

A triangular plate of iron weighing 5 cwt. is carried horizontally by 3 men, one at each angular point; find the weight supported by each.

6. Find the pressure at any point in a mass of fluid at rest, having given that the pressure of the air at the surface of the fluid is P.

A town is supplied with water from a reservoir, the level of the water in which is 136 feet above the lowest point of the main supplypipe; assuming that the height of the Water-Barometer is 34 feet, and that the pressure of air on a square inch is 15 pounds, required the pressure which the main must be constructed to bear.

7. Describe the Common Hydrometer; and shew how to compare the Specific Gravities of two fluids by means of it.

An Hydrometer, used for determining the Specific Gravity of spirits, is constructed in such a way that the zero point of graduation (at the top of the stem) is on the level of the fluid, when it is placed in proof spirit, and that the interval between two successive graduations corresponds to 10th part of its bulk. If the reading of the instrument, when placed in spirit sold as proof, be 27, determine the amount of water unfairly introduced (Specific Gravity of pure alcohol =0'8).

8. Describe an experiment for proving that the elastic force of air at a given temperature varies as the density.

A Barometer contains a small quantity of air above the mercury; having given that the length of the tube measured from the constant level of the cistern is 32 inches, and that the mercury in it stands at 29.5 inches, when a standard Barometer is at 30, obtain a formula for obtaining the true height of the Barometer from the observed height generally.

9. Describe the construction of the Common Air-Pump, and its operation.

By what contrivance may the degree of exhaustion of the air in the Receiver be exhibited to the eye? Why cannot a perfect vacuum be produced by means of this instrument?

10. Explain the method of filling, and graduating, a Common Mercurial Thermometer.

How may the height of a mountain be roughly determined by means of the Thermometer?

FIRST DIVISION.—B.

1. Define the terms “Force”, and “Weight”.

Explain clearly the method of estimating and comparing statical forces.

2. If two weights, acting perpendicularly on a straight Lever on opposite sides of the fulcrum, balance each other, they are inversely as their distance from the fulcrum.

A tobacconist buys tobacco wholesale at the rate of 3s. 4d. a pound, and in weighing it out to his customers uses a Balance, the arms of which are as 16 to 15; if he profess to sell it at 3s. 9d. a pound, what profit per cent. does he really make?

3. Enunciate the 'parallelogram of forces'; and assuming it to be true so far as the magnitude of the Resultant is concerned, complete the proof.

It is found that if, on a rapid river, a ferry-boat be turned obliquely to the stream and prevented going down the stream by means of chains stretched across from one bank to the other, it will be carried across by the force of the stream alone. this.

Explain

4. Investigate the condition of equilibrium when a weight W is supported on an Inclined Plane by a force P acting parallel to the plane.

If the force P be the tension of a fine thread, which passes over a small fixed pulley, and is attached to a weight hanging freely, shew that if P be pulled down through a given space, the height of the Centre of Gravity of P and W will remain unaltered.

5. When a body is placed on a horizontal plane, it will stand or fall, according as the vertical line through its Centre of Gravity falls within or without the 'base'.

Why is the word horizontal introduced in the enunciation of this proposition? With what modifications is it true of any plane?

6. Shew that the pressure on the horizontal bottom of a vessel, filled with fluid, depends merely on its depth below the surface, and not at all on the quantity of fluid contained in it.

A pipe carries rain water from the top of a house to a large tank, the surplus water in which escapes through a valve in the top which rises freely. A weight of 21 lbs. is placed on it, and it is found that the water rises in the pipe to the height of 20 feet before the valve opens. Required its area (assuming that the height of the Water-Barometer is 34 feet, and the atmospheric pressure 15 lbs. on a square inch).

7. Describe the Hydrostatic Balance; and shew how it may be applied to compare the Specific Gravities of two fluids, by weighing the same solid in each.

A piece of copper of Specific Gravity 885 weighs 887 grains in water, and 910 grains in alcohol; required the Specific Gravity of the alcohol.

8. Describe the construction of the Condenser, and the mode of its operation.

If the volume of the cylinder be one-fifth the volume of the Receiver, find the pressure at any point of the latter after 20 strokes.

9. Explain the construction of the Common Mercurial Baro

meter.

Having given that the Specific Gravity of mercury is 13:57, and that the weight of a cubic inch of water is 2526 grains, find the pressure of the air on a square inch in lbs., when the mercury in the Barometer stands at 30.5 inches.

10.

Describe the construction of the Common Pump.

If the upward movement of the piston be stopped, when the water has risen to a given height (say 16 feet) in the supply-pipe, but has not yet reached the piston, find the tension of the piston-rod (the area of the piston being 4 square inches, and the atmospheric pressure on a square inch being known).