The time of oscillation t depends upon (i) d, (ii) r, and (iii) *.
i.e., t ∝ da rb sc
or, t = k da rb sc ......(iii)
Where k is the constant of proportionality, (dimensionless).
Writing the dimensional formulae of the various physical quantities in equation (i):
Dimensional formula for 't' = [M0L0T1]
Dimensional formula for 'd' = [ML-3T0]
Dimensional formula for 'r' = [M0L1T0]
Dimensional formula for '*' = [M1L0T-2]
Therefore, equation (i) becomes,
[M0L0T] = [M1L-3T0]a [M0L1T0]b [M1L0T-2]a
= [Ma+c L-3a +b T-2c]
Using the principle of homogeneity:
a + c = 0 ........(ii)
-3a +b = 0 ....(iii)
-2c = 1 ........(iv)
Form equation (iv), c = -1/2
Form equation (ii) a = -c = -(-1/2) = 1/2
Form equation (iii) b = 3a =3 x 1/2 = 3/2
Substituting the values of a, b and c in equation (i) we get,
