Assume that the displacement(s) of air is proportional to the pressure difference (Δp) created by a sound wave. Displacement (s) further depends on the speed of sound (v), density of air (ρ) and the frequency (f). If Δp 10 Pa, v 300 m/s, ρ 1 kg/m³ and f 1000 Hz, then s will be of the order of (take the multiplicative constant to be 1) :

Solution

Physics sir · 4 steps

Write s as a product of powers of Δp, v, ρ and f, then match M, L and T on both sides.

1. Set up powers. Since s ∝ Δ p, take s = Δ p vᵃ ρᵇ fᶜ. Pressure has dimensions [M L⁻¹ T⁻²], v is [L T⁻¹], ρ is [M L⁻³], f is [T⁻¹], and s is [L].

   [L] = [M L⁻¹ T⁻²] [L T⁻¹]ᵃ [M L⁻³]ᵇ [T⁻¹]ᶜ

2. Match dimensions. Mass: 1 + b = 0, so b = -1. Length: -1 + a - 3b = 1, so a = -1. Time: -2 - a - c = 0, so c = -1. That gives s = (Δ p)/(ρ v f).
3. Put in numbers. s = 10/(1 × 300 × 1000) = 1/30000 m = 3.3 × 10⁻⁵ m, which is about 0.033 mm.
4. So. s ≈ 1/30 mm, which is of the order of 3/100 mm, option (b).

The idea to keep. When a quantity depends on several others, write it as a product of their powers and equate the exponents of M, L and T. Three base dimensions let you solve for three unknown powers.

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