Question

Two strings (fixed at both ends) $P$ and $Q$ of equal length, made of same material and vibrating in same mode produces beats of frequency $5\text{Hz}$. Initial string $Q$ is vibrating at a frequency $460\text{Hz}$. If the tension in string $P$ increases, the beat frequency increases to $7\text{Hz}$. The initial frequency of vibration for string $P$ is :

• A. $455\text{Hz}$
• B. $465\text{Hz}$
• C. $453\text{Hz}$
• D. $462\text{Hz}$

Answer 1

The correct option is B $465\text{Hz}$

Initial frequency of string $Q$ is $460\text{Hz}$
Given the beat frequency for string $P$ and $Q$,
$f_B=5\text{Hz}$
$\Rightarrow |f_Q-f_P|=5$
$\Rightarrow 460-f_P=\pm 5$
Therefore the initial frequency of $P$ is either $455\text{Hz}$ or $465\text{Hz}$.

Since, both the strings(fixed at both ends) are made of same material and vibrating in same mode, hence $\lambda$ and $\mu$ remains fixed.
$v=f\lambda$
$\Rightarrow v\propto f...\left(i\right)$
and $v=\sqrt{\frac{T}{\mu }}$
$\Rightarrow v\propto T...\left(ii\right)$
From Eq.$\left(i\right),\left(ii\right)$ we get:
$f\propto T$
It clearly indicates that increase in tension in string $P$ will lead to increase in its vibrating frequency.
Also after $T_P\uparrow$,
$f_B=7\text{Hz}$
$\Rightarrow f_Q-f_P^{\prime }=\pm 7$
$\Rightarrow f_P^{\prime }=460\pm 7$
$\therefore f_P^{\prime }=453\text{Hz}\text{Or}467\text{Hz}$
Hence initial frequency of string must be $465\text{Hz}$, so that it increases to $467\text{Hz}$ after increasing tension.