Question

If tension is increased by $4\%$ in vibrating string, then percentage change in speed of wave will be

• A. $1\%$
• B. $2\%$
• C. $3\%$
• D. $4\%$

Answer 1

The correct option is B $2\%$

Given:
$\frac{dT}{T}\times 100=4\%$

We know that, speed of wave,
$v=\sqrt{\frac{T}{\mu }}$
where $T$ is tension and $\mu$ is mass per unit length.

Taking natural $\log$ both sides, we get

$\ln v=\ln \left(\sqrt{\frac{T}{\mu }}\right)$

$\Rightarrow \ln v=\frac{1}{2}\ln \left(\frac{T}{\mu }\right)$

$\Rightarrow \ln v=\frac{1}{2}(\ln T-\ln \mu )$

On differentiating both side,

$\frac{1}{v}dv=\frac{1}{2}(\frac{1}{T}dT-0)[\mu =\text{constant}]$

$\Rightarrow \frac{dv}{v}=\frac{dT}{2T}$

For percentage change in speed and tension,

$\Rightarrow \frac{dv}{v}\times 100=\frac{dT}{2T}\times 100$

$\Rightarrow \frac{dv}{v}\times 100=\frac{1}{2}\times 4=2\%$

So, percentage change in speed of wave is $2\%$

Hence, option $\left(b\right)$ is the correct answer.