diminished by the resistance of the water). Also find the average force in grammes-weight exerted on the bullet by the powder (8=981 centimetres per second, per second). Int. Sc. 1885. 169. Show that the difference of kinetic energy of two particles, after sliding through the same distance down two planes, equally inclined to the horizon, but one rough, the other smooth, is numerically equal to the work done against friction by the former particle. Owens Coll. 1884. 170. Prove that train going at 45 miles an hour will be brought to rest in about 284 yards by the breaks, supposing them to press on the wheels with half the weight of the train, and that the coefficient of friction is ·16. Int. Sc. 1885. 171. A cannon-ball weighing 10,000 grammes is discharged with a velocity of 45,000 centimetres per second from a cannon, the length of whose barrel is 200 centimetres : prove that the mean force exerted on the ball during the explosion is 5.0625 x 1010 dynes. Camb. Schol. 1884. 172. Calculate the amount of work done against gravity in drawing a car of 2.5 tons weight, laden with thirty passengers averaging 9 stones each in weight, up an incline the ends of which differ 120 feet in level. Find the horse-power sufficient to do that work in half an hour. 173. A string has one end fastened to a fixed point; it then goes under a pulley which sustains a weight of 2 kilogs., and then over a fixed pulley, and at its end it sustains another weight of 2 kilogs. The strings are all vertical, and the weight of the pulleys and all friction may be neglected. The weight is pulled up by the free weight descending. Find how long it will take thus to raise the weight through 2 metres, and show that 392,000,000 ergs more work can be done by the kinetic energy of the system. Balliol Coll. 1880. 174. The amount of work which can be derived from Edinb. M.A. 1882. ounce the consumption of a kilogramme of coke is 309 x 1012 ergs. Of the work thus derived one-twentieth is usefully employed in drawing trucks up a slope of 30°. Find the amount of coke required to draw a weight of 9400 megadynes a distance of 1500 metres along the slope. (1 megadyne = 106 dynes.) Ind. C. S. 1885. 175. Define kinetic energy and work. Calculate the kinetic energy of a tram-car weighing 2.5 tons, when it is moving at the rate of 6 miles an hour, and is laden with thirty-six passengers, averaging 9 stones each in weight. If the coefficient of kinetic friction for a tramcar moving on its rails is į ; find how much work is done when the above car, loaded as stated, is pulled 3 miles along a level road. Edinb. M.A. 1881. 176. Compare the amounts of momentum and of kinetic energy in (a) a pillow of 20 lbs., which has fallen through one foot vertically, and (6) an bullet moving at 200 feet per second. Edinb. M.A. 1881. 177. Compare the strength of a locomotive engine that can get up a speed of 20 miles an hour in 2 minutes in a train of 350 tons with that of an engine that can get up a speed of 30 miles an hour in 2 minutes in a train of 250 tons, the line in both cases being level, and friction negligible. Univ. Coll. Lond. 1884. 178. Show that if a body falls freely from rest under the action of gravity, the increase of its kinetic energy during the roth second is 9.5 g times the increase of momentum during that second, where g is the numerical value of gravity. Camb. Schol. 1883. 179. An engine of one horse-power is capable of doing 33,000 foot-pounds of work per minute. What is the horse-power of an engine which can pump 1000 gallons of water per minute from a well and project it with a velocity of 80 feet per second through a nozzle which is at a height of 40 feet above the surface of the water in the well ? Int. Sc. 1882. 180. A railway train of 300 tons is running at 45 miles an hour. Find its energy of motion in foot-tons. Also, if the engine, while getting up the speed of the train, does work on it at an average rate of 100 horsepower, show how long it will take to get up the speed of 45 miles an hour. [Take g=32 (feet, seconds); I horse-power = 33,000 foot-pounds per minute.] Int. Sc. 1884. 181. What is the horse-power of an engine which can project 10,000 lbs. of water per minute, with a velocity of 80 feet per second, 20 per cent of the whole work done being wasted by friction, etc. ? [N.B.--An agent of one horse-power can do 33,000 foot-pounds of work per minute.] Matric. 1884. 182. If a bicyclist always works at the rate of onetenth of a horse-power, and goes 12 miles an hour on the level, prove that the resistance of the road is 3.125 pounds. Int. Sc. 1885. 183. A railway train of mass 100 tons is moving at 20 miles per hour. What horse-power would be required to impart to it this velocity in 5 minutes from starting, in addition to overcoming the resistances supposed uniform and equal to 12 lbs. per ton ? (One horse-power = 33,000 foot-pounds per minute.) Camb. Schol. 1884. CHAPTER II HYDROSTATICS IN solving the examples in this chapter, the following facts may be assumed : The mass of a cubic foot of water is 1000 ounces or 621 pounds. The specific gravity of mercury is 13.6. In examples on fluid pressure, the pressure of the atmosphere may be neglected, unless the contrary is expressly stated. The normal atmospheric pressure is that due to a column of mercury 76 centimetres in height. When the inch is taken as the unit of length, the normal barometric height may be assumed as 30 inches. Geometrical Relations. The ratio of the circumference of a circle to its diameter is 3.1416 (or approximately 22), and is denoted by the Greek letter 1. The circumference of a circle of radius r is 2nr, and its area is ar2. The area of the surface of a sphere is 45r2. 1. Define a fluid, and explain what is meant by the pressure of a fluid at any point within it. Prove that the pressure at any point of a fluid at rest is the same in every direction. 2. State and illustrate the law of transmission of pressures in liquids, and explain how it is applied in the construction of hydraulic presses. 3. Distinguish between pressure and intensity of pressure ; and find the dimensions of each of these quantities in terms of the fundamental units of length, time, and mass. 4. Define density and specific gravity, pointing out the essential distinction between them. Explain how it is that the density of a substance has, in the C.G.S. system, the same numerical value as its specific gravity, or density relative to water. 5. A block of mahogany 2 inches long, ii inch broad, and inch thick is found to weigh 1 461 grains : express its density in grains per cubic inch. The volume of the block is 2x*x=41 cubic inches, and its mass is 461 grains. The density, in grains per cubic inch, is 21 8 6. A ton of clay is found to occupy a volume of 18 cubic feet : what is its density in pounds per cubic foot ? 7. The density of water is 621 pounds per cubic foot: express this in ounces per cubic inch. 8. Basalt is three times as heavy as water : what is its density in ounces per cubic inch? 9. A ton of chalk occupies a volume of 151 cubic feet : what is its specific gravity referred to water as the standard substance ? The mass of a cubic foot of the chalk is (2240: 91) pounds, and the mass of a cubic foot of water is 62} pounds. Hence the required specific gravity (or ratio between the masses of equal volumes) is 2 _125_2240 X 2 X 2 2240 x = 2:312. 2 31 X 125 10. A cubic inch of a substance weighs half a pound: what is its specific gravity ? * In this and the succeeding chapters the term weight is occasionally used instead of mass, when there is no danger of confusion between the two. When a body is said to be weighed in air, the weight of the displaced air may be neglected, unless the contrary is expressly stated. |