Question

Given that time –period (t) of a soap bubble depends upon its radius (r), density(ρ) and surface tension( T ). The dimensionally correct relation is
a) t =K r^1/2ρ1/2

b) T t=K r^3/2 ρ^1/2 T 1/2

c) t= Kr^1/2ρT

d) t=Kr3/2 ρ ^ -1/2 T^ -1/2​

Answer 1

Let time period be related to given quantities as follows:

$t \propto {r}^{x} \: { \rho}^{y} \: {T}^{z}$

• Expressing these quantities in terms of Mass (M) , length (L) and time (T).

$\implies T \propto {L}^{x} \: { (M{L}^{ - 3} )}^{y} \: {(ML{T}^{ - 2} )}^{z}$

$\implies T \propto ( {M}^{y + z} ) \: ( {L}^{x - 3y + z} ) \: ( {T}^{ - 2z} )$

• After comparing the sides, the 3 equations we get are:

$1) \: y + z = 0$

$2) \: x - 3y + z = 0$

$3) \: - 2z = 1$

• After solving the equations, we get:

$1) \: x = 2$

$2) \: y = \dfrac{1}{2}$

$3) \: z = - \dfrac{1}{2}$

So, final equation becomes :

$\boxed{t = k \bigg( {r}^{2} \: { \rho}^{ \frac{1}{2} } \: {T}^{ \frac{ - 1}{2} } \bigg)}$