2. At a bend in a river, the velocity in a certain part of the surface is 170 centims. per second, and the radius of curvature of the lines of flow is 9100 centims. Find the slope of the surface in a section transverse to the lines of flow.

Ans. Here the centrifugal force for a gramme of the (170)2

water is

=3176 dynes. If g be 981 the slope will

; that is, the surface will slope upwards

from the concave side at a gradient of 1 in 309.

The

general rule applicable to questions of this kind is that the resultant of centrifugal force and gravity must be normal to the surface.

3. An open vessel of liquid is made to rotate rapidly round a vertical axis. Find the number of revolutions that must be made per minute in order to obtain a slope of 30° at a part of the surface distant 10 centims. from the axis, the value of g being 981.

f

Ans. We must have tan 30°

=

where ƒ denotes the

" g

intensity of centrifugal force—that is, the centrifugal force

per unit mass. We have therefore

4. For the intensity of centrifugal force at the equator due to the earth's rotation, we have rearth's radius 638 x 10, T=86164, being the number of seconds in a sidereal day.

=

If the earth were at rest, the value of g at the equator would be greater than at present by this amount.

If the

earth were revolving about 17 times as fast as at present, the value of g at the equator would be nil.

30

CHAPTER IV.

HYDROSTATICS.

33. THE following table of the relative density of water at various temperatures (under atmospheric pressure), the density at 4° C. being taken as unity, is from Rossetti's results deduced from all the best experiments (Ann. Ch. Phys. x. 461; xvii. 370, 1869):—

34. According to Kupffer's observations, as reduced by Professor W. H. Miller, the absolute density (in grammes per cubic centimetre) at 4° is not 1, but

1'000013. Multiplying the above numbers by this factor, we obtain the following table of absolute densities:

Temp. Density. Temp. Density. Temp.

Density.

35. The volume, at temperature t°, of the water which occupies unit volume at 4°, is approximately

I + A (t-4)2-B (t−4)26 + C (t−4)3,

and the relative density at temperature t° is given by the same formula with the signs of A, B, and C reversed. The rate of expansion at temperature ť° is

2A (1 − ́4) — 2·6B (t− 4)1·6 + 3C (t − 4)2.

In determining the signs of the terms with the fractional exponents 2.6 and 16, these exponents are to be regarded as odd.

36. Table of Densities (chiefly taken from Rankine's "Rules and Tables," pp. 149 and 150):—

37. If a body weighs m grammes in vacuo and m' grammes in water of density unity, the volume of the body is m-m' cubic centims.; for the mass of the water displaced is m‐m' grammes, and each gramme of this water occupies a cubic centimetre.

Examples.

1. A glass cylinder, / centims. long, weighs m grammes in vacuo and m' grammes in water of unit density. its radius.

Find