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to be moving with a velocity of 10 metres per second: how many ergs of work have been done upon it?

113. How many foot-pounds of work must be done on a mass of one ton in order to give it a velocity of 15 miles an hour?

114. The mass of a pendulum-bob is 100 grammes, and the string is a metre long. The bob is held so that the string is horizontal, and is then allowed to fall: find its kinetic energy when the string makes an angle of 30° with the vertical.

115. A shot travelling at the rate of 200 metres per second is just able to pierce a plank 4 cm. thick: what velocity is required to pierce a plank 12 cm. thick?

Assuming that the resistance offered by the plank is uniform, it follows from the equation

Fs=mv2

that the thickness which the shot can penetrate is proportional to the square of its velocity. If a shot moving with velocity can pierce a plank of thickness t, and a shot moving with velocity v' can pierce a plank of thickness t', then

In the above example

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v2=(200)2 × 12/4 = 120,000,

and therefore the required velocity is 346.4 metres per second.

116. If a bullet moving with a velocity of 150 metres per second can penetrate 2 cm. into a block of wood, through what distance would it penetrate when moving at the rate of 450 metres per second?

117. What is the energy of a train of 40 tons moving at the rate of 30 miles an hour? What force, acting for 30 seconds, would be sufficient to bring the train to rest?

118. A stone of mass 3 lbs. falls from rest for 2 seconds, when it comes in contact with a flat roofingslate, which it smashes, thereby losing two-thirds of its

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velocity: how much energy does it lose by breaking the slate?

119. Compare the amount of kinetic energy in (1) a boulder of one hundredweight which has fallen for one second from rest, and (2) a one-pound projectile moving with a velocity of 800 feet per second.

120. A bullet of 90 grammes leaves the muzzle of a gun with a velocity of 500 metres per second. If the barrel be 120 centimetres long, find the mean pressure exerted by the powder upon the bullet.

121. A railway carriage contains forty passengers, whose average weight is 140 lbs. If the carriage itself weighs 6 tons, and is moving at a rate of 30 miles an hour, what is its kinetic energy?

122. A body of mass m moves under the action of a force F through a space s in a straight line, which is inclined at an angle to the direction of the force: if v be the velocity generated, show that Fs cos 0 = 1⁄2mv2.

123. A 20-lb. cannon ball falls through a vertical distance of 1600 feet: what is its energy? With what velocity would it have to be projected from a cannon in order to possess an equal amount of energy y?

124. Two inelastic balls moving in opposite directions come into collision; the one has a mass 10 and velocity 50, the other a mass 50 and velocity 10: what is their total kinetic energy before and after impact, and what has become of the energy apparently destroyed?

125. A body of mass 56 lbs. starts from rest under the action of a constant force, and acquires a velocity of 64 feet per second while moving through a space of 160 feet find the acting force and the work done by it.

126. A constant force acts upon a body for 20 seconds, doing 10 units of work upon it, and generating in it during the same time 30 units of momentum : find the mass of the body and the velocity which it will have acquired.

127. The bob of a simple pendulum is let go when the

pendulum is inclined at an angle of 60° to the vertical. Compare its kinetic energy after describing an arc of 30° with its K.E. at its lowest point.

Power. The power (or activity) of an agent is the rate at which it can do work, and is measured by the number of units of work done per unit of time. The unit of power commonly used by engineers in this country is the horse-power, which is defined as being the power of doing 33,000 foot-pounds of work per minute, or 550 foot-pounds of work per second.

128. Assuming that the pressure within the cylinder of a steam-engine remains constant throughout the whole of the stroke, find the horse-power developed in each cylinder of an engine, having given—

A = area of piston in square inches.

P = pressure upon the piston in pounds per square inch. S = length of stroke in feet.

R = number of revolutions per minute.

Here P denotes the intensity of the pressure on the piston in pounds weight per square inch. The total pressure on the piston is the weight of AP pounds. This is the acting force, and. the distance through which it moves in each stroke is S feet.

Thus the work done in each stroke is SAP foot-pounds. Since there are two strokes for each revolution, the number of strokes per minute is 2R, and the work done per minute is 2SRAP foot-pounds. Thus the horse-power developed

is

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129. Water is supplied to a hydraulic motor at a pressure of 100 lbs. per square inch. Express the potential energy of the water in the reservoir in footpounds per gallon; and calculate the maximum H.P. which can be developed by the motor if the rate of supply is 50 gallons per minute.

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height in feet of the reservoir.

The pressure in

pounds weight per square foot = hp, where p = number of pounds in a cubic foot of water = 62.5,

.. h× 62.5/144 = pressure in pounds weight per square inch = 100,

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Thus the potential energy of one pound of water is 230.4 footpounds, and the potential energy of one gallon (or 10 lbs.) is 2304 foot-pounds.

(Notice that since the pressure is supposed constant in the question, we must also assume that the level of the water in the reservoir is kept constant.)

The work done by a supply of 50 gallons per minute is

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130. The nominal value of a horse-power is 33,000 foot-pounds per minute. Express this (1) in kilogrammemetres per minute and (2) in ergs per second.

131. A five H. P. engine is employed to pump water from the bottom of a mine 100 feet deep.

How many cubic feet of water will it raise in 24 hours? (I cub. ft. of water

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62 lbs.)

132. What should be the indicated H.P. of an engine that is intended to pump 200 gallons of water per minute to a height of 50 yards? (1 gal. = 10 lbs.)

133. A 300 H.P. engine draws a train of 180 tons, the resistance due to friction being 12 lbs. per ton find its maximum velocity along a level line.

134. Find the H.P. of an engine that should be employed for raising coal from a pit 200 feet deep, the average daily yield being 1782 tons.

135. Find the horse-power exerted by an engine which draws 150 tons up an incline of 1 in 200 at the rate of 15 miles per hour, the resistance due to friction, etc. being equivalent to a weight of 14 lbs. for every ton in motion.

136. Determine the rate at which an engine is working when it drives a train of 150 tons at a rate of 30 miles an hour, the resistance to motion being equal to a weight of 16 lbs. for every ton.

137. The mass of a train is 200 tons, and the resistances to its motion amount to 20 lbs. per ton on a level line find the horse-power of an engine which can just keep it going at the rate of 45 miles an hour.

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138. A steam-engine supplies 1000 houses with 100 gallons of water each, working 12 hours per day: if the mean height to which the water has to be raised is 80 feet, at what rate does the engine work?

139. What alteration would be produced in the unit of work if the units of mass, length, and time were each increased ten-fold? If a horse-power be represented by 550 under the old system, what would be its numerical value in the new?

EXAMINATION QUESTIONS.1

140. A particle whose mass is M pounds moves from rest under the action of a force of P units which is constant in magnitude and direction: how far will the particle move in ʼn seconds, and what space will it describe in the nth second?

If the force be the weight of the body, and the

1 The following abbreviations are used in marking the sources from which the examples at the end of each chapter are taken :

London University

Examinations

Camb. Schol.

Camb. B.A.

N. S. Tripos

M. Tripos

Balliol Coll.

Vict. Int.

Ind. C. S.

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Cambridge Inter-Collegiate Scholarship Examinations.
Cambridge General and Special B.A. Exams.

Natural Science Tripos.

Mathematical Tripos.

Brakenbury Natural Science Scholarship, Balliol College.
Intermediate B.A. and B.Sc. Exams., Victoria University.
Indian Civil Service.

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