Question

A rectangular pulse of width 0.5 cm is travelling to the right on a taut string (shown by full line in the figure) that has mass per unit length 𝜇1. The string is attached to another taut string (shown by dashed line) of mass per unit length 𝜇2. If the tension in both the strings is the same, and the transmitted pulse has width 0.7 cm, the ratio 𝜇1⁄𝜇2 is _______(rounded off to two decimal places)

Answer 1

Correct Answer: 1.94 - 1.98

Given: 
Width of incident pulse: \( w_1 = 0.5\ \text{cm} \)
Width of transmitted pulse: \( w_2 = 0.7\ \text{cm} \)
Tension in both strings is the same.

Concept:
Time duration of the pulse remains the same at the boundary. So, \[ \frac{w_2}{w_1} = \frac{v_2}{v_1} \] Wave speed on a string: \[ v = \sqrt{\frac{T}{\mu}} \] Thus, \[ \frac{v_2}{v_1} = \sqrt{\frac{\mu_1}{\mu_2}} \] Therefore, \[ \frac{w_2}{w_1} = \sqrt{\frac{\mu_1}{\mu_2}} \] Calculation:
\[ \sqrt{\frac{\mu_1}{\mu_2}} = \frac{0.7}{0.5} = 1.4 \] Squaring, \[ \frac{\mu_1}{\mu_2} = (1.4)^2 = 1.96 \] Final Answer:
\[ \frac{\mu_1}{\mu_2} = 1.96 \]