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Find the resultant, by construction or otherwise, of forces of 5 and 15 units acting at a point and inclined at 120°. 5. What is meant in mechanics by a couple and its moment ? Mention some familiar instances of couples. How would you make it clear that a couple cannot be kept in equilibrium by a single force ? 6. What is the centre of a system of parallel forces ? If two weights of 3 lb. and 7 lb. balance on a lever, one of whose arms is 5 in. longer than the other, find the lengths of both the arms. 7. Find the force that will keep a body at rest on an inclined plane when the force is (i.) horizontal and (ii.) parallel to the plane. Two unequal weights, connected by a string, are in equilibrium on two smooth inclined planes, the string passing over a pulley at the intersection of the planes : find the ratio between the weights. 8. Distinguish between a solid and a fluid, between a liquid and a gas, and between density and specific gravity. What is the special character of a perfect fluid ? The specific gravity of brass being taken as 8, and the mass of one cubic foot of water as 1.000 oz., find the density of brass in ounces to the cubic inch. 9. Describe the essential features of the mercurial barometer, and explain why the mercury falls when carried up a mountain. When carried up 1,000 ft. will the mercury fall ten times as much as it does when carried up 100 ft. ? Why ? 10. Show how you would prove, experimentally or otherwise, that the pressure at any point in a heavy liquid at rest is proportional to the depth below the free surface of the liquid. Find the pressure on a square inch of the base of a cone one foot high, standing with its vertex upwards, and just filled with water. 11. Two liquids which do not mix meet in a bent tube: show how to compare their densities by observing the relative heights of the three surfaces.

No. 40. — Theoretical Mechanics. — For Class C and for Civil Service Senior. Time allowed: Three hours. [All answers should be illustrated, where possible, with diagrams.] 1. What do you understand by relative velocity ? State and justify the theorem known as the " parallelogram of velocities." For what other magnitudes can a similar theorem be stated? A train travels 100 miles in 3 hours 50 seconds: what is its average velocity in feet a second ? 2. Distinguish carefully between kinetic and potential energy, and show that the kinetic energy of a mass m moving with velocity v is \ mv*. A train weighing 500 tons runs on a level road against a total resistance of 101b. a ton: how many units of work must be expended in a run of 50 miles? 3. Explain the principle and describe the applications of the simple pendulum. What other varieties of pendulum are there ? A simple pendulum, supposed to beat seconds, loses 20 seconds in a day : find by how much its length should be altered to make it accurate. i. What are the complete conditions that three forces should be in equilibrium ? Show that forces represented completely by the three medians AD, BE, OF of a triangle ABC will be in equilibrium. 5. Show, with proof, how to find the resultant of two like parallel forces. Parallel forces of 5, 4, 3 units act at points 4, 6, 10 inches respectively from one end of a line: find the magnitude and position of the resultant. 6. Where are the centres of gravity of a triangle, a pyramid, and a cone? Establish the case of the triangle. The mass of an equilateral triangle is 61b., and masses of lib., lib., and 21b. respectively are placed at its angular points : find the centre of mass of the system. 7. A salesman's balance has unequal arms a, b, and he weighs goods alternately in one scale and the other to the same nominal amount in each : does he gain or lose by his balance not being true, and how much? 8. State and prove a rule for finding the whole pressure of a liquid on a body, or the magnitude of the resultant pressure on a plane area, immersed in it. A sphere, whose radius is 2 ft., has its centre 5 ft. below the surface of water in which it is immersed : find the whole pressure and also the resultant pressure of the water on it. 9. Describe the hydrometer of variable immersion. A hydrometer of this kind is put first into a liquid whose S.G. is 0 - 80, and then into a liquid whose S.G. is 0 - 90, a mark being made in each case at the point of the stem that is at the surface of the liquid, and one mark is found to be 6 in. from the other: how far above or below the surface will each of the marks be when the hydrometer floats in a liquid whose S.G. is 0-87 ? 10. Describe the force pump and its action. If the cross-section of the piston has an area of 8 square inches, with what pressure must it be forced down to raise water to an elevation of 120 ft.

No. 41. — Physiography. — For Class C. Time allowed: Three hours. {Candidates are requested to answer only eight questions. Diagrams should be used wherever applicable.] 1. Account for — (a.) The phases of the moon. (b.) An eclipse of the moon, (c.) The same face of the moon always being presented to the earth.