The percentage increase in the speed of transverse waves produced in a stretched string if the tension is increased by 4%, will be __________%.

Solution

Physics sir · 4 steps

Speed goes as the square root of tension, so a small percentage change in tension gets halved.

1. Wave speed. For a stretched string with tension T and mass per unit length μ, the speed of a transverse wave is set by these two alone. The string is the same, so μ does not change.

   v = √(T/μ) ⇒ (Δ v)/v × 100 = 1/2((Δ T)/T × 100)

2. Put in numbers. The tension rises by 4%, so the speed rises by 1/2 × 4% = 2%.
3. Exact check. Without the approximation, v₂/v₁ = √1.04 ≈ 1.0198. That is a rise of about 1.98%, which agrees with 2%.
4. So. The speed of the transverse wave increases by 2%.

The common slip: 4. Treating v as directly proportional to T gives 4, not 2. Speed depends on the square root of tension, so halve the percentage.

The idea to keep. For y ∝ xⁿ, a small percentage change in x gives n times that percentage change in y. With v ∝ √T, take half the change in tension.

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