power of the length, and as the nth power of g, and to be independent of everything else, the dimensions of a time. must equal the mth power of a length, multiplied by the nth power of an acceleration, that is T= L" (LT-2)" = L" L" T-2n = Lm+nT-2n Since the dimensions of both members are to be identical, we have, by equating the exponents of T, that is, the time of vibration varies directly as the square root of the length, and inversely as the square root of g. 2. The velocity of sound in a gas depends only on the density D of the gas and its coefficient of elasticity E, and we shall assume it to vary as Dm E". The dimensions of velocity are LT-1. The dimensions of density, or mass volume' are ML-3. The dimensions of E, which will be explained in the force chapter on stress and strain, are or (MLT-2)L-2, or area ML-1 T-2. The equation of dimensions is LT-1 Mm L-3 = = · Mm+n L-3m―n T-2n, whence, by equating coefficients, we have the three equations 1 = -3m - n, - 1 = - 2n, m+n= 0, to determine the two unknowns m and n. and these values will be found to satisfy the first equation also. The velocity, then, varies directly as the square root of E, and inversely as the square root of D. 3. The frequency of vibration f for a musical string (that is, the number of vibrations per unit time) depends on its length 7, its mass m, and the force with which it is stretched F. The dimensions of ƒ are T-1. 4. The angular acceleration of a uniform disc round its axis depends on the applied couple G, the mass of the disc M, and its radius R. Assume it to vary as G* M" R2. The dimensions of angular acceleration are T-2. giving whence 2. = 2x, x+y=0, 2x + z = x=1, y=-1, 2= 2. Hence the angular acceleration varies as = 0, G MR2 In the following example the information obtained is less complete :— 5. The range of a projectile on a horizontal plane through the point of projection depends on the initial velocity V, the intensity of gravity g, and the angle of elevation a. The dimensions of range are L. inferences as to the manner in which a enters the expression for the range. The dimensions of this expression will depend upon V and g alone. Assume that the range varies as Vm g". Then CHAPTER IV. HYDROSTATICS. 34. 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) :— 35. 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 den 36. The volume, at temperature t', of the water which occupies unit volume at 4°, is approximately 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 t° is 2A(t-4)-2.6B(t - 4)16 x 3C(t-4)2. In determining the signs of the terms with the fractional exponents 2.6 and 1.6, these exponents are to be regarded as odd. 37. The following Table of Densities has been compiled by collating the best authorities, but is only to be taken |