Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is v₁, and that in the other string having radius of cross section R/2 is v₂. Then v₂/v₁ =

Solution

Physics sir · 4 steps

Same tension, same material: the wave speed depends only on the radius, as v ∝ 1/r.

1. Speed on a string. A transverse wave on a string travels at v = √(T/μ), where μ is the mass per unit length. Tension T is the same for both strings.
2. Mass per length. For a string of density ρ and radius r, μ = ρ π r². The material is the same, so ρ is the same and μ ∝ r².

   v = √(T/(ρ π r²)) ∝ 1/r

3. Take the ratio. v₂/v₁ = r₁/r₂ = R/(R/2) = 2.
4. So. The thinner string carries the wave twice as fast, so v₂/v₁ = 2, which is option (c).

The idea to keep. For strings under equal tension, v ∝ 1/√μ. Since μ ∝ r², the speed goes as 1/r. Always check whether the square root cancels the square.

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