Question

Two mechanical waves on strings of equal length (\( L \)) and tension (\( T \)) having linear mass density \( \mu_1/\mu_2 = 1/2 \). Find the ratio of time taken for a wave pulse to travel from one end to the other in both strings. (Ignore gravity)

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

Hint:

The velocity of a wave on a string depends on the tension and mass density of the string. For strings of equal length and tension, the ratio of the times taken is the inverse square root of the ratio of the mass densities.

Answer 1

The Correct Option is B

To solve this problem, we need to determine the time taken for a wave pulse to travel along two different strings. The critical formula here is the speed of a wave on a string, given by:

\[ v = \sqrt{\frac{T}{\mu}} \]

where \( v \) is the wave speed, \( T \) is the tension in the string, and \( \mu \) is the linear mass density of the string.

Now, let's denote the linear mass density of the first string as \( \mu_1 \) and the second string as \( \mu_2 \), and it is given that \(\frac{\mu_1}{\mu_2} = \frac{1}{2}\).

Therefore, we can write:

\[ \mu_2 = 2 \mu_1 \]

 

Let's determine the speeds of the wave on both strings:

1. For the first string:

\[ v_1 = \sqrt{\frac{T}{\mu_1}} \]

1. For the second string:

\[ v_2 = \sqrt{\frac{T}{\mu_2}} = \sqrt{\frac{T}{2\mu_1}} \]

Substitute \(\mu_2 = 2 \mu_1\) into the expression for \(v_2\), we have:

\[ v_2 = \frac{1}{\sqrt{2}} \cdot \sqrt{\frac{T}{\mu_1}} = \frac{v_1}{\sqrt{2}} \]

Now, the time taken for a wave pulse to travel a length \( L \) is:

\[ t = \frac{L}{v} \]

Thus, the time taken for the wave pulse to travel the length on both strings is:

1. Time on the first string:

\[ t_1 = \frac{L}{v_1} \]

1. Time on the second string:

\[ t_2 = \frac{L}{v_2} = \frac{L}{v_1/\sqrt{2}} = \frac{L \cdot \sqrt{2}}{v_1} \]

The ratio of the time taken for a wave pulse to travel through the first and the second string is:

\[ \frac{t_1}{t_2} = \frac{\frac{L}{v_1}}{\frac{L \cdot \sqrt{2}}{v_1}} = \frac{1}{\sqrt{2}} \]

Hence, the correct answer is \( \frac{1}{\sqrt{2}} \).