A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation T of the liquid drop depends on the radius ( r ) of the drop, density ρ and surface tension S of the liquid. Which among the following will be a possible expression for the time period? where k is a dimensionless constant.

Question

A spherical liquid drop is placed on a horizontal plane. A small disturbance causes the volume of the drop to oscillate. The time period of oscillation T of the liquid drop depends on the radius ( r ) of the drop, density ρ and surface tension S of the liquid. Which among the following will be a possible expression for the time period? where k is a dimensionless constant., k ρ r S, k ρ 2 r S, k ρ r 3 S, k ρ r 3 S 2

MARKS App · Accepted Answer

According to the question, time period, T ∝ r a ρ b S c T = k r a ρ b S c ..... (i) Thus, putting dimension, we get [ T ] = [ L ] a [ M L − 3 ] b [ M T − 2 ] c T = M b + c ⋅ L a - 3 b ⋅ T - 2 c Equating the dimensions of both sides, we get b + c = 0 , a - 3 b = 0 and - 2 c = 1 ∴ c = - 1 2 ∴ b = 1 2 and a = 3 b = 3 2 Putting these value of a , b and c into Equation. (i), we get T = k ⋅ r 3 2 ρ 1 2 ⋅ S - 1 2 = k ρ S r 3

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