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gation. We shall describe two methods, one of which will always be applicable if the substance is solid and insoluble in water. The method would have to be modified if the substance were soluble in water, or if it were hygroscopic.

Method I. If a body is weighed first in air, and then suspended in a liquid, its apparent weight will be less in the second case than in the first, and it has been known since the time of Archimedes, that the apparent loss of weight is equal to the weight of the liquid displaced by the body. If M is the mass of the body, and p its density, its volume is M/p, and if ☛ is the density of the fluid in which it is weighed, the apparent decrease of mass will be

Mo
ρ

The same holds for a weighing in air, the density λ of air being substituted for σ.

Assuming the arms of the balance to be a and b cms. respectively, we have for the moments about the central knifeedge, of the forces on the two arms during the weighing in air, the quantities

a Mg (1-2)

and bM,g (1-1),

where M, is the apparent mass, and σ, the density of the weights. For equilibrium these moments must be equal, hence

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Similarly for the weighing in water, if M, is the apparent mass

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Dividing the first equation by the difference between the first and second, we have

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i.e., the excess of density of body over that of air

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× (excess of density of liquid over that of air). It will be noticed that neither the inequality of the arms of the balance, nor the buoyancy effect of the air on the brass weights, enters into the result, so long as the weights are always placed in the same pan. This is due to density determinations depending on ratios of weights only.

It will also be seen that the equation giving p corrected for the buoyancy of the air, may be obtained from the equation

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in which the air is neglected, by subtracting the density of the air from each density occurring in the equation. This may be seen on consideration to be due to the fact, that the weight of a body obtained in air would be the same as the weight obtained in vacuo, if the density of the body as it was transferred from air to vacuo, were decreased by the density of the air. This holds for both the quartz and the water in the above case, and the principle will be used in other cases.

The numerical calculation is best carried out by writing 1- for σ, and p' for M1/(M,- M1) when we have

p=p' - p' (x +λ) + λ,

where the last two terms are small.

Exercise I. Determination of the density of Quartz by weighing in water.

1. Place a wooden stool across the left-hand pan, place upon it an empty beaker of suitable size. Estimate the length I between the hook at top of the balance pan and the centre of the beaker, and take two pieces of silk about 20 cms. longer than this estimated length, use one to tie round the quartz crystal provided, leaving a length with a loop at the end, hanging from the crystal. Cut away all unnecessary thread, and cut off equal lengths from the other piece.

2. Suspend the quartz by means of the thread from the hook underneath the top of the support of the left-hand balance pan, place the other piece of thread in the right-hand pan, and find the weights required to produce equilibrium.

3. Remove the quartz from the balance, place it in water in a beaker, and boil the water to drive off the air bubbles adhering to the quartz. Then cool the water by pouring into it water from the tap gently without causing splashes. Place a small wooden stool across the left-hand pan of the balance, so that the pan does not come into contact with it at any point. Support the beaker on this stool and suspend the quartz again from the hook. See that the quartz is entirely immersed in the water and that no bubbles of air adhere to it. Place a thermometer in the water, and note the temperature (Fig. 16). Cut off from the piece of thread in the right-hand pan a length. equal to the length of thread in the water, and remove it. Weigh the quartz.

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Temperature of Water, 20° C.

Apparent Weight of Quartz in Water (M2)

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40.882 grms.

25.487

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= 2.6556

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Hence p the density of the quartz at 20°C. = 2·6491.

Method II. If the solid can only be obtained in small pieces,

we may determine the density by the use of a

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'specific gravity flask" (Fig. 19), which is a small glass flask provided with a well-ground stopper traversed by a narrow channel. When it is filled with a liquid, and the stopper is inserted carefully so as to exclude air bubbles, the excess of liquid will flow out through the capillary opening. By means of a piece of blotting paper a small quantity of the liquid may be removed, so that it just reaches to a marked height in the opening. The flask may in this way be repeatedly filled to the same level, and if its temperature is the same, the volume of its contents will be the same.

Fig. 19.

The flask having been cleaned and dried, the density required is determined by the following series of weighings, the letters, F, &c., representing the weights obtained:

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2. The flask dry with the dry solid placed inside, F + M1.

3.

4.

The flask with the solid inside, after filling up to the mark with a liquid of known density σ at a temperature t1, F+ M2+ W1.

The flask entirely filled up to the mark with a liquid of density σ, at a temperature t2, F+ W2.

Since σ is the density of the liquid at t2, the volume of the flask at t2, neglecting the effect of the air on the weighing, is

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W 2

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W2 1+ at where a is the coefficient σ2 1+ at2'

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is the volume occupied by the solid at t1, and the density p of the solid at t, is the mass divided by this volume.

The effect of the buoyancy of the air may be taken into

account by subtracting the density of air λ, from each of the densities in the equation for p (page 55), and we thus get

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If we take weighings of the flask empty and when filled with each of two liquids, we may compare the densities of the liquids, since on making M1 = 0 in the above equation, we have

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an equation from which we can determine one density if the

other is known.

If in the former equation, we take the liquid to be water in each case, so that σ and σ are both nearly unity, we have, using the methods of approximation given in Intermediate Practical Physics (page 16),

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Exercise II. Determination of the density of Quartz by the specific gravity flask method.

1. Clean the 50 gram flask (Fig. 19) provided, by washing it if necessary with a strong solution of caustic potash, and then with water from the tap. The potash is to be thoroughly removed with tap water, and the final washing made with distilled water.

Place the flask on the shelf of an air bath kept at about 120° C., insert a glass tube into the flask, through an opening in the top of the bath. Look at the flask occasionally, and when the drops of water have evaporated from the sides, draw the moist air out of the flask, by applying the mouth to the

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