specific gravity and the density of a substance are identical.

A nugget of gold mixed with quartz weighs 12 ounces, and has a specific gravity 6.4; given that the specific gravity of gold is 19.35, and of quartz is 2.15, find (to one place of decimals) the quantity of gold in the nugget.

B.Sc. 1891.

192. A thin uniform rod six feet long of specific gravity 75 floats partly immersed in water, its upper end being supported by a string 3 feet long whose other end is attached to a hook 4 feet above the surface of the water. Find the magnitudes and directions of the forces acting on the stick and the angle the stick makes with the surface of the water. Draw a diagram to scale showing the position of equilibrium of the system. Camb. Schol. 1891.

193. Explain how Archimedes' principle enables us to compare the densities of liquids and solids.

A piece of lead weighs 7.88 grammes in air, 7.19 in water, and 7.33 in alcohol; a piece of oak weighs 13.21 grammes in air, and the oak and lead together weigh 4.87 grammes in water: find the specifie gravities of lead, oak, and alcohol. Camb. Schol. 1893.

194. A balloon is filled with a gas whose specific gravity is th of that of air at the pressure of 760 mm. of mercury at o° C. Compare the lifting power of the balloon in air when the height of the barometer is 750 mm, with its lifting power when the barometer stands at 760 mm. The temperature in both cases is o° C., and the volume of the balloon is supposed to remain unaltered. Matric. 1894.

195. A common hydrometer whose weight is 100 grammes floats immersed up to a certain point on its stem in a liquid whose specific gravity is .9. What weight must be attached to the top of the stem in order that the hydrometer when floating in water may be

immersed up to the same depth? Will the stability of the hydrometer be the same in the two cases? Camb. Schol. 1891. 196. State Boyle's Law, and explain the term “height of the homogeneous atmosphere."

An accurate barometer reads 30 inches when one containing air above the mercury reads 24 inches. If the tube of the latter be raised 3 inches, the reading becomes 25 inches. Find what length of the tube the air would occupy if brought to atmospheric pressure. Vict. Int. Sc. 1890.

197. A flask when empty weighs 120 grammes, when full of air it weighs 121.3 grammes, and when full of water 1220 grammes: calculate the density of the air. Explain whether it is or is not necessary to take account of the weight of air displaced.

Matric. 1891.

198. What do you know about the density of gases in relation to temperature and pressure? Describe experiments which show that the density of a gas at constant temperature is proportional to its pressure. A uniform tube closed at top, open at bottom, is plunged into mercury, so that it contains 25 c.c. of gas at atmospheric pressure 76 cm.: the tube is now raised until the gas occupies 50 c.c.: how much has it been raised? Matric. 1891.

199. Explain the construction of a barometer, what it measures, and how it measures it. Translate pressure measured in terms of the height of a barometer mercury column (say either 27 inches or 60 centimetres) into absolute units of pressure. Matric. 1891.

200. A glass bulb will just stand an excess of inside over outside pressure of 200 gm. per sq. cm. It is sealed up at a place where the barometer stands at 75 cm., and then taken uphill till it bursts. What is the height of the barometer at the place where this occurs?

Matric. 1895.

201. A certain quantity of a gas has a volume of 5

cubic feet, and its pressure is equal to that which is due to 28 inches of mercury; a certain quantity of another gas has a volume of 4 cubic feet, and its pressure is equal to that which is due to 25 inches of mercury. When they are mixed their volume is 8 cubic feet; find from first principles the pressure of the mixture, the temperature of the gases and of the mixture being the

same.

Find

S. K. 1895. 202. A barometer reads 30 inches at the base of a tower, and 29.8 inches at the top, 180 feet above. the average mass of a cubic foot of air taking the specific gravity of mercury as mass of a cubic foot of water as 62.4 lbs.

in the tower, 13.5, and the

Matric. 1894.

203. When the barometer is standing at 30 inches, a uniform straight tube, closed at one end, is partly filled with mercury and held in a vertical position with its closed end upwards and with its open end in a tank of mercury. When the closed end is 25 inches above the surface of the mercury in the tank the surface of the mercury inside the tube is one inch from the closed end. How far must the tube be pulled up in order that the top of the mercury inside it may be 6 inches from the closed end? Camb. B.A. 1891.

204. What must be the least diameter given to a soap-bubble inflated with hydrogen gas so that it may just float in air; the temperature being 15° C. and the pressure 770 mm.; the thickness of the soap-bubble film being 002 cm. ?

[Density of mercury, 13.596 at 0° C. Density (at o°C and under a pressure of 1 megadyne) of air .0012752: of hydrogen 00008837 grammes per c.c.]

Camb. Schol. 1891

CHAPTER IV

HEAT-EXPANSION

1. Expansion of Solids

Definition. The coefficient of linear expansion of a substance may be defined in any of the following ways:

1. If a bar of a substance be heated through one degree, its length will increase by a certain fraction, and this fraction is called the coefficient of linear expansion of the substance.

2. The coefficient of linear expansion is the ratio of the increase of length produced by a rise of 1° to the original lengths. 3. The coefficient of linear expansion is numerically equal to the increase of length produced in unit length of the substance by a rise in temperature of 1°.

Let a denote the coefficient of linear expansion of a body whose length at o° is : starting with any one of the above definitions you will easily see

(1) that the expansion produced by heating the body through 1° is la,

(2) that the expansion produced by heating it to t is at, and hence

(3) that its length l at t° is given by the equation

or, - (coeff. of exp.) =

total expansion

(original length) × (interval of temperature)

No solid or liquid expands with absolute uniformity, but the statements above made may be regarded as approximately

accurate.

In the case of a body which expands irregularly we may take equation (2) as defining its mean coefficient of expansion. In all three definitions given above it would be more strictly accurate to use the expression "from o° to 1°."

APPROXIMATE COEFFICIENTS OF LINEAR EXPANSION.

These values may be assumed in solving the examples in this chapter.

1. Find the length at 200° of a zinc rod whose length at o° is 128 cm.

If the length at 200° be denoted by 1200, then

2. A piece of brass wire is exactly 3 metres long at 250°: what will be its length at o°?

Using the same system of notation, we have, by equation (1), 10=1250/(1+250a)

= 300/(1 + 250 × 0.000019)

= 300/1.00475=298.582 cm.

3. The distance between two marks on a brass bar

is found to be 90 cm. at 10°.

between the marks at 90° ?

What will be the distance

The expansion of the brass on heating through 80° will be lat=90 x 0.000019 × 80=0.1368 cm., and hence the distance between the marks at 90° will be 90.1368 cm.

Note. The solution here given assumes that if 4 be the length of a bar at any temperature t°, its length / at any other temperature '° is