The frequency (ν) of an oscillating liquid drop may depend upon radius (r) of the drop, density (ρ) of liquid and the surface tension (s) of the liquid as ν=rᵃρᵇ sᶜ. The values of a, b and c respectively are

Solution

Physics sir · 3 steps

Write every quantity in M, L, T, then match the powers on both sides; time alone fixes c first.

1. Write dimensions. Frequency is [T⁻¹]. Radius is [L], density is [ML⁻³], and surface tension (force per length) is [MT⁻²].

   M⁰L⁰T⁻¹ = Lᵃ(ML⁻³)ᵇ(MT⁻²)ᶜ = Mᵇ⁺ᶜ Lᵃ⁻³ᵇ T⁻²ᶜ

2. Match the powers. Time: -2c = -1, so c = 1/2. Mass: b + c = 0, so b = -1/2. Length: a - 3b = 0, so a = 3b = -3/2.
3. So. (a, b, c) = (-3/2, -1/2, 1/2), giving ν ∝ √(s/(ρ r³)). That is option (b).

The idea to keep. In dimensional analysis, start with the dimension that appears in only one quantity. It gives one exponent directly, and the rest follow one by one.

Answer from the NTA final answer key. Solution by PrepWiser.