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2. Two bodies, A and B, describe circles with constant velocities; the radius of A's circle is 390 times that of B's; A moves round its circle once while B moves round its circle 13 times; A's mass is 91 times B's mass. Compare the force acting on A with the force acting on B.

3. A mass of 20 lbs. is revolving uniformly, once in 5 seconds, in a circle whose radius is 3 feet. Find the centrifugal force.

wheel 2 ft. in diameter, which The linear velocity of the rim

4. A mass of 1 lb. is placed on the rim of a revolves upon its axis, and is otherwise balanced. being 30 ft. per sec., what is the pull on the axis as caused by the mass of 1 lb. 5. A locomotive of 20 tons runs at the rate of 30 miles an hour at a part of the line where the radius of curvature is 10 miles; calculate in tons its entire pressure against the inner surfaces of the rails.

6. A locomotive, 15 tons in weight, runs with a velocity of 20 miles an hour in a circle of a mile radius; calculate in poundals its entire pressure against the rails.

7. A railway carriage weighing 4 tons is passing round a curve, the radius of which is 250 yards, at the rate of 20 miles an hour; what is the outward pressure on the rails?

8. If the earth were set to revolve on its axis in half a day, nothing else being altered, what would be the weight of a mass of 100 lbs., tested by a spring balance, supposing the balance to have been graduated to show pounds' weight at the equator, when the earth was revolving with its actual angular velocity? Earth's radius is 21 × 106 feet; g=32'09 at the equator, with actual angular velocity.

9. A railway carriage is going round a curve of 500 feet radius, at 30 miles per hour. Find how much a plummet hung from the roof by a thread 6 feet long would be deflected from the vertical.

10. A skater describes a circle of 100 feet radius, with a velocity of 18 feet per second; what is his inclination to the ice?

11. Find the force per gramme of the earth's mass towards the sun, supposing the earth's motion to be in a circle of radius 1'473 × 1013 centimetres.

12. Calculate the tension of an endless chain, of 1 pound per foot, and 30 feet long, when made to rotate as a horizontal circle once per second.

13. Masses of 6 and 16 M are fixed at the ends of a horizontal rod; round what vertical axis must it revolve that the two centrifugal forces may destroy one another; first when the rod is supposed without mass, second when its mass is 4 M.

14. What ought to be the difference of level between the rails, when the radius of curvature of a railway curve is 300 yards, the breadth of the guage 4 ft. Sin., and the highest velocity of a train at the place 45 miles an hour?

15. A square frame-work of a yard in the side rotates once per second about

one of its sides, which is vertical; what must be the coefficient of friction to keep a ring from slipping down the opposite side?

16. A weight of 28 lbs. is suspended by means of a ring on a straight rod which revolves in a horizontal plane about a fixed axis; the coefficient of friction between the ring and the rod is 1, and the distance of the weight from the axis is 5 ft.: find the angular velocity when the weight will begin to move outwards.

17. A mass of 2 pounds is kept performing simple harmonic vibrations, at the rate of 30 periods per second, through a range of a tenth of a foot on each side. Find the maximum force, and the force at 1/50 foot from the middle position.

18. A mass of 1/1000 pound vibrates 256 times in a second through a range of 1/10 inch. Find the maximum force upon it.

19. A mass of a gramme vibrates through a millimetre on each side of its middle position 256 times per second; find the maximum force upon it in grammes weight. Assume the intensity of gravity at 9814 centimetres per

second per second.

20. A body whose mass is 10 lbs. is tied to a thread 6 ft. long, and is allowed to swing backwards and forwards through the arc of a semi-circle; when it is 30° from the lowest point of the arc, what forces are acting on it?

21. A seconds pendulum is lengthened and the time of oscillation is thereby increased by an eighth of a second. Calculate the increase in length. (g = 322.) 22. Given that the intensity of gravity at Paris is 9'81 metres per second; what is the length of the pendulum which beats seconds there.

23. A pendulum, 39 20 inches long, vibrates seconds of mean time at a certain place. Find the force of gravity per pound of matter at that place. How much faster would the pendulum beat at a place where the force of gravity is a quarter per cent. greater?

24. What would be the length of a simple pendulum vibrating in 2.5 seconds, at a place where the intensity of gravity is 31.5 ft. per sec. per sec.

25. What must be the length of a pendulum beating 10 times in a minute if the seconds pendulum is 39 inches long.

26. A pendulum 37 8 inches long is found to make 182 beats in 3 minutes at a certain place; find the force of gravity at the place.

27. At a place where a simple pendulum 100 centimetres long beats seconds, find how many seconds are gained per day when the pendulum is shortened by one millimetre.

28. A clock, whose pendulum ought to beat seconds, is gaining at the rate of hour per week. How many turns should be given to the screw-head, supposing it to have 40 turns to the inch, to correct the error in length of the pendulum. Take the seconds pendulum at 393 inches.

SECTION XXX.-SPECIFIC GRAVITY.

ART. 149. Heaviness and Density. We have

1

F=M

F = M by L per T per T.

The acceleration at a particular place due to the attraction of the earth is the same for every kind of matter; let it be denoted by L per T per T, then

g F=M.

Let the density of a substance be p M=V, by eliminating M we deduce

pg F = V.

This gives us the idea of weight per volume, that is, of heaviness.

ART. 150.-Specific Gravity. By the specific gravity of a substance is meant its heaviness compared with the heaviness of a standard substance, as water at the temperature of 62° Fahr. (British), or at its temperature of maximum density (French). As the intensity of gravity is the same for all kinds of matter, the specific gravity, that is, the relative heaviness, has the same value as the specific mass, that is, the relative density.

ART. 151.-Buoyancy. Let the heaviness of a solid be

and of a fluid

pg F = V,

p'g F=V.

When the solid is immersed in the fluid, the heaviness of the fluid acts as an upward force, and the resulting heaviness of the solid is

i.e.

pg - p'g F = V,
(p-p')g F=V.

When p' is greater than P, the resulting heaviness becomes negative, that is, it becomes buoyancy.

The determination of the density of water at 4° C. upon which the kilogramme was based, was effected by weighing a brass Rankine's Rules and Tables, p. 102

cylinder in water. The cylinder used is represented in the accompanying illustration. It was hollow but completely closed,

[graphic]

except that a small tube kept up the connection between the air in the cylinder and the air in the room when the cylinder was immersed in water. The ratio of hollow to solid cylinder was so arranged that the weight of the metal was only a little greater than the buoyancy of the water displaced. The diameter and height were each intended to be 2-435 decimetres. By means of the lines drawn on its surface the mean diameter was found at 17° C. to be 2-428368 dm., and the mean height 2.437672 dm. The volume of the cylinder at 0° C. was computed to be 11-28 cubic decimetres, and that of the solid part 1.506 cb. dm. The weighing was effected by means of a provisional kilogramme and

M

its decimal parts. Let the provisional kilogramme be denoted by kilo; then the final results of the weighings at 60° C. were as

follows:

Weight of the cylinder in air,

weight of the cylinder in distilled water,

therefore weight of water having the volume of the cylinder,

therefore

i.e. Corrected for the

11.4660055 kilo,

0.1967668 kilo ;

11.2692387 kilo;

11-2692387 kilo = 11.28 cubic decimetres,
999046 kilo

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- cubic decimetre.

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and the primary kilogramme was constructed to equal

•999207 kilo.

EXAMPLES.

Ex. 1. How much would a brass kilogramme appear to weigh if it were suspended in water?

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g dynes = cc. of water,

(831) g dynes apparent = 8.3g dynes real,

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Observation.―g dynes = gm., hence gm. may be substituted in the above instead of g dynes.

Ex. 2. A solid weighs 100 grains in water and 120 grains in alcohol of specific gravity 0.8. Find the mass and the specific gravity of the solid.

Suppose that the mass of the solid is a grains. Then (x-100) grains is its apparent loss of mass in water, ... (100) grains is

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