this principle is true for each portion of the motion, and therefore throughout the whole.

Now the hammer starts from rest, and finally comes to rest. Hence the whole work done upon it must be zero. But it falls altogether through 5 feet under a constant force tons. Hence the work

equal to the weight of 25 (1+377)

όπ

14/

done upon it by this downward force is 125 (1+ foot tons. Therefore the work which the hammer must do upon

1

foot tons. But it compresses the

iron through of a foot. Hence the pressure which it exerts

12

upon the iron must be equal to the weight of

1. A particle is projected from the vertex of a smooth parabolic tube, whose axis is vertical, and latus rectum equal to 4a, along the tube with a velocity represented by √2ag. Find the velocity of the particle at any point in the tube in terms of the focal distance of the point.

2. Assuming that on descending a mine, g varies directly as the distance from the earth's centre, find the number of beats lost in a day by a pendulum which beats seconds at the sea-level, when carried down a mine to a depth of 400 fathoms, supposing the earth a sphere of 4000 miles radius.

3. Shew that the time of oscillation of a particle under the action of gravity about the lowest point of a vertical circle of radius 2a, is greater than the time of oscillation on a cycloid the diameter of whose generating circle is a, if the arc of oscillation in the circle be of finite length.

4. A particle slides down the surface of a right circular cylinder whose axis is horizontal from rest on the highest generating line. Find the pressure on the cylinder in any subsequent position of the particle, and the point where the particle will leave the surface.

5. Supposing the bob of a conical pendulum to weigh 20 lbs., and the length of the string to be 3 feet, find the inclination of the string to the vertical when the bob is making 3 revolutions per second, and its tension.

6. A heavy particle is attached to a string 5 feet long, and swung round in a vertical circle. Find its velocity at the highest point in order that the string may just remain tight.

7. Explain the action of the conical pendulum as a regulator or "governor" for a steam-engine.

8. In the case proposed in question 1, find the pressure of the particle upon the tube at any point of its path.

9. A train goes round a curve whose radius (i. e. the radius of the curve lying midway between the two metals) is 150 yards, at the rate of 50 miles per hour. Find the height to which one of the metals must be raised above the other in order that the whole pressure of each carriage on the metals may be perpendicular to the floor of the carriage, the breadth of the gauge being 4 ft. 8 ins.

10. A heavy particle is suspended from the angular points of an equilateral triangle whose plane is horizontal, by means of three strings each equal in length to one side of the triangle. If one of the strings be cut, find the initial change of tension of the other two.

11. A uniform string, of length 27a, is rotating in its own plane with uniform velocity w, under the action of no external forces. Find the tension of the string, the mass of each unit of length being m.

12. A smooth wedge whose angle is 30° weighs 10 lbs., and rests on a smooth plane inclined 30° to the horizon, so

that the upper surface of the wedge is horizontal. A weight of 2 lbs. is placed on the top of the wedge. Find its acceleration and the pressure of the weight on the wedge.

EXAMPLES ON CHAPTER V.

1. The value of g at Greenwich being 321912 and at Trinidad 32 0913, find how many beats a Greenwich seconds' pendulum would lose in a day at Trinidad.

2. Shew that the acceleration of a particle oscillating in a smooth cycloidal tube whose axis is vertical, is at any point proportional to its distance from the vertex measured along the curve.

3. Find the inclination to the vertical of a conical pendulum 20 inches long, and making 200 rotations per minute.

4. A body weighing 10 lbs. is suspended by a string from a point in the roof of a railway carriage, which is describing a curve of 509 feet radius at the rate of 45 miles an hour. Find the inclination of the string to the vertical when it is in relative equilibrium, and the tension of the string.

5. Find the difference in the pressures exerted on the metals by a train weighing 200 tons when going due East, and when going due West, along a horizontal rail at 60 miles an hour in latitude 60°.

6. A pendulum which at A beats seconds, gains 2 beats an hour at B. Compare the weights of the same substance at the different places.

7. Two very small imperfectly elastic balls are let fall simultaneously from different points, their centres moving on the same cycloid whose axis is vertical and vertex downwards. Shew that all their impacts will take place at the vertex, and find the ultimate range of vibration when the impacts have ceased.

8. The length of a pendulum which vibrates 30 times in a minute is 1568 inches, find the space through which a particle will fall from rest in one second under the action of gravity.

9. A free body falls from rest through nearly 3013 yards in one-eighth of a minute in the latitude of Greenwich. How far would a body fall from rest in a quarter of a minute at a place where the length of the seconds' pendulum is 999 of its length at Greenwich?

m

10. The attraction of a planet of mass m on a given body at a point distant from its centre, r being greater than the radius of the planet, varies as The mass of the earth is 49 times that of a certain planet, while its radius is 4 times that of the planet. Prove that a seconds' pen

7

dulum carried to the planet would oscillate in about 4

seconds.

11. If a simple pendulum 39 inches long oscillate in one second, what is the length of a pendulum which makes 3540 beats in an hour?

12. The horizontal attraction of a mountain on a particle at a certain place is such as would produce in it an acceleration denoted by g. Shew that a seconds' pendulum

at that place will gain

1

η

13. Shew that a pendulum one mile long would oscillate

14. A seconds' pendulum is carried to the top of a mountain 3000 feet high; assuming that the force of gravity varies inversely as the square of the distance from the earth's centre, and that the earth's radius is 4000 miles, find the

number of oscillations lost in a day, neglecting the attraction of the mountain.

15. A railway train is moving uniformly along a curve at the rate of 60 miles per hour, and in one of the carriages a pendulum which would ordinarily beat seconds, is observed to oscillate 121 times in two minutes. Shew that the radius of the curve is very nearly a quarter of a mile.

Supposing a stone dropped from the window of one of the carriages, find how much farther from the centre of the curve is the point at which it strikes the ground than the point vertically beneath that from which it falls, the height of the latter point above the ground being 6 feet.

16. A particle is projected horizontally with a given velocity from the highest point of a smooth sphere. Find the point where it leaves the sphere.

17. Find the greatest velocity with which a particle may be projected horizontally from the highest point of a sphere, so as to move on the surface of a sphere.

18. A smooth straight tube is made to describe a right circular cone whose axis is vertical and semivertical angle equal to a, with uniform velocity, the vertical plane through the tube turning about the axis of the cone with uniform velocity w. Find where a particle will be in relative equilibrium in the tube.

19. Shew that if a heavy particle fall from a cusp down the arc of a smooth cycloid whose axis is vertical and vertex downwards, its pressure on the curve at its lowest point will be equal to twice its weight.

20. Supposing the earth's orbit about the sun to be a circle of 86,000,000 miles radius, and the earth to describe this orbit with uniform velocity in 365 days, express the force exerted by the sun on a pound of matter at the earth's surface in British absolute units, neglecting the magnitude of the earth in comparison with the sun's distance.