First slide

Beats

Question

The ends of two identical strings are tied to two rigid walls and same tension is maintained in them. The strings are oscillating in their fundamental made with frequency 100 Hz. If the tension in one of them is increased by 6%, then frequency of beats is

Moderate

A

3
B

2
C

4
D

5

Solution

Fundamental frequency $f = \frac{1}{2 l} \sqrt{\frac{T}{μ}}$
When tension in one string is increased by 6%,
$\begin{array}{l} f^{1} = \frac{1}{2 l} \sqrt{\frac{T^{1}}{μ}} \\ ∴ \frac{f^{1}}{f} = \sqrt{\frac{T^{1}}{T}} = \sqrt{\frac{1 .06 T}{T}} = \sqrt{1 + \frac{6}{100}} = 1 + \frac{1}{2} \frac{6}{100} \\ ∴ \frac{f^{1} - f}{f} = \frac{1}{2} \frac{6}{100} \Rightarrow f^{1} - f = \frac{6}{2 \times 100} \times 100 = 3 \end{array}$

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