EXAMPLES 185 



GENERAL EXAMPLES 



1. Prove that the horse power of an engine which overcomes a resist- 

 ance of R pounds at a speed of S miles an hour is 



RS -4- 375. 



2. A train weighing, with the locomotive, 500 tons is kept moving at 

 the uniform rate of 30 miles an hour on the level, the resistance of air, 

 friction, etc. , being 40 pounds per ton. Find the horse power of the engine. 



By how much must this horse power be increased if the rate is to be 

 maintained while water is taken up from a trough between the rails to the 

 amount of 20 pounds per foot passed over, the height to which the water is 

 raised above the trough being 10 feet, and the kinetic energy imparted to 

 the water in the trough, as well as that of the motion of the water taken up, 

 relatively to the tank, being neglected ? 



3. The sides of a conical hill are of such a shape that a given mass will 

 just rest on them without slipping. A man wishes to move this mass from 

 a point at the base of the hill to a second point diametrically opposite to 

 the first. Show that the work of dragging it over the hill is less than the 

 work of dragging it round the base of the hill, in the ratio 2 : TT. 



4. Show that the work a man does in dragging a weight up a hill from 

 a given point A to the summit B depends only on the positions of A and 

 B, and is independent of the shape of the hill, provided he keeps always 

 in the vertical plane through A and B. 



5. A catapult is made by tying the two ends of a piece of elastic, natural 

 length a, modulus of elasticity X, to the two prongs of a forked piece of 

 wood, distant I apart, I being greater than a. A stone of mass m is placed 

 at the middle point of the catapult, and is drawn back until the string is 

 stretched to double its natural length. If it is then set free, find the 

 velocity with which it will leave the catapult. 



6. If , in the last question, the stone is projected vertically upwards from 

 the catapult, find the height to which it will rise before coming to rest. 



7. A necklace of mass m is made of beads threaded on a light string of 

 modulus X. It is held in a horizontal plane, with the string unstretched, 

 resting on the surface of a smooth right circular cone of semivertical angle 

 a, of which the axis is vertical. If the necklace is let go, how far will it 

 slip down the cone before coming to rest ? 



8. A fly wheel is of radius 2 feet 6 inches, and the weight of the spokes, 

 etc., may be neglected in comparison with that of the rim. It is rotating 

 at the rate of 250 revolutions per minute about a fixed axle, which is 

 3 inches in diameter, the coefficient of friction between the wheel and axle 

 being ^. If it is left to itself, find how many revolutions it will make 

 before stopping. 



