move gently to their position of equilibrium, show the line joining the nail to the axle will make an a of 30° with the vertical.

Also, if the weights be, instead, allowed to m freely after the nailing, find the kinetic energy of weights and wheels when the nail arrives at this p tion.

154. If two weights w and 2w are connected b string passing over a smooth weightless pulley, wh is attached to a third weight 3w by a string pass over a smooth fixed pulley, prove that the weight descends with an acceleration

Camb. Schol. 1881

Let a denote the acceleration of the weight 2w downwa relative to the pulley P, and let B note the acceleration of P upwards. of the weight 3w downwards. The actual acceleration of 2w in space relative to the fixed pulley P') is a Let be the tension in the string c necting w and 2w. Considering motion of 2w, we see that the fo causing motion is 2wg-7, and the m moved is 2w, and the acceleration duced is a - B.

assuming 2= 10, g= 32 feet per second at the pole, show that the weight of a body will be decreased by about of the whole by taking it from the pole to the equator.

Balliol Coll. 1881.

156. What is erroneous in the term centrifugal force? A skater describes a circle of radius 100 feet with a velocity of 18 feet per second: what is his inclination to the ice? Edinb. M.A. 1884.

157. Investigate the relation between the length of a simple pendulum and the time of oscillation. A "second's" pendulum is lengthened by 1 per cent : how much will it lose in a day? (g=32.)

Glasgow M.A. 1882. 158. What is meant by a "simple pendulum," and "equivalent simple pendulum "?

A pendulum vibrates seconds at a certain place; it is taken to another where the value of g is greater by I per cent.: find the number of vibrations greater or less per day. Glasgow M.A. 1884. 159. A pendulum 37.8 inches long makes 183 beats in 3 minutes at a certain place: find the force of gravity there.

Ind. C. S. 1886.

160. What is the meaning of the expression "the length of the seconds - pendulum"? How far could it be rational to make that length, or some sub-multiple of it, the unit of length? What would be the acceleration of gravity if the length of the seconds - pendulum were the unit of length? N. S. Tripos. 1884.

161. A balloon ascends with a constant acceleration, and reaches a height of 900 feet in one minute. Show that a pendulum clock carried with it will gain at the rate of 27.8 seconds per hour. M. Tripos. 1883. 162. Explain the terms mass of a pound, weight of a pound, foot-pound.

A mass of iron weighing 20 pounds (1) falls freely under the action of gravity (g= 32 ft.-sec.); (2) descends acceleration of 25 (ft. - sec.) Determine in

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each case the kinetic energy of the mass in foot-pou when it has fallen through 100 feet.

163. A man weighing 12 stones does 360,000 f pounds of work against gravity in ascending a h find the height ascended. Glasgow M.A. 188 164. Masses of 485 and 496 grammes respectiv are connected by a fine string and hung over a v light pulley as in an Atwood machine. Find, by c sidering the work done, the velocity after a fall of 88 from rest, supposing that friction may be neglect (g=981.)

If the actual velocity is only three-fourths the calcula velocity, find the energy which has been taken up by friction. Vict. Int. 1885

165. State the laws of friction.

A curling rink is 42 yards long from tee to tee. curling stone is hurled from one tee with a velocity 8.4 feet per second, and comes to rest at the other t Find the coefficient of friction between the stone a the ice. Edinb. M.A. 1884

166. A constant force acting on a body generates 10 seconds 20 units of momentum, and does in 1 same time 20 units of work: find the mass of 1 body and its velocity at the end of 10 seconds.

diminished by the resistance of the water). Also find the average force in grammes-weight exerted on the bullet by the powder (g=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 a 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.

Edinb. M.A. 1882.

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 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.

weight descending.

Balliol Coll. 1880.

the consumption of a kilogramme of coke is 309 x I Of the work thus derived one-twentieth is usefu ergs. employed in drawing trucks up a slope of 30°. amount of coke required to draw a weight of 94 megadynes a distance of 1500 metres along the slo Ind. C. S. 1885 (1 megadyne = 106 dynes.)

Calcul

175. Define kinetic energy and work. the kinetic energy of a tram-car weighing 2.5 to when it is moving at the rate of 6 miles an hour, and laden with thirty-six passengers, averaging 9 stones ea in weight.

If the coefficient of kinetic friction for a tra car moving on its rails is ; find how much work done when the above car, loaded as stated, is pulled miles along a level road.

Edinb. M.A. 1881.

176. Compare the amounts of momentum and kinetic energy in (a) a pillow of 20 lbs., which fallen through one foot vertically, and (b) an bullet moving at 200 feet per second.

our

Edinb. M.A. 1881.

177. Compare the strength of a locomotive engi that can get up a speed of 20 miles an hour in 2 minu in a train of 350 tons with that of an engine that c get up a speed of 30 miles an hour in 2 minutes in train of 250 tons, the line in both cases being level, a Univ. Coll. Lond. 1884. friction negligible.

178. Show that if a body falls freely from rest und the action of gravity, the increase of its kinetic ener during the 10th second is 9.5 g times the increase momentum during that second, where g is the numeric value of gravity.

179. An engine of one horse-power is capable doing 33,000 foot-pounds of work per minute. What the horse-power of an engine which can pump 100 gallons of water per minute from a well and project with a velocity of 80 feet per second through a nozz which is at a height of 40 feet above the surface of th water in the well?