Question

Two mechanical waves travel on strings of equal length \(L\) and equal tension \(T\). The linear mass densities of the strings are in the ratio \(\dfrac{\mu_1}{\mu_2} = \dfrac{1}{2}\). Find the ratio of time taken for a wave pulse to travel from one end to the other in both strings. (Neglect gravity).

• A. \(\dfrac{1}{2}\)
• B. \(\dfrac{1}{\sqrt{2}}\)
• C. \(\sqrt{2}\)
• D. \(2\)

Hint:

For waves on strings:
Speed increases with tension
Speed decreases with linear mass density
Time is inversely proportional to wave speed

Answer 1

The Correct Option is B

Wave speed on a stretched string is given by: \[ v = \sqrt{\frac{T}{\mu}} \] Time taken to travel length \(L\): \[ t = \frac{L}{v} = L\sqrt{\frac{\mu}{T}} \] Step 1: Ratio of times: \[ \frac{t_1}{t_2} = \sqrt{\frac{\mu_1}{\mu_2}} \]
Step 2: Substitute given ratio: \[ \frac{t_1}{t_2} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]