Question

A wave is propagating on a long stretched string. If the cross-sectional area of string is doubled then average power associated with wave becomes

• A. twice of its initial value
• B. half of its initial value
• C. four times of its initial value
• D. none of these

Answer 1

The correct option is A twice of its initial value

We know that average power transmitted in string is,
$P_{\text{avg}}=2{\pi }^2{f}^2\mu v{A}^2$
where mass per unit length is,
$\mu =\frac{m}{l}=\frac{s\times l\times \rho }{l}=\rho s$
$\Rightarrow P_{\text{avg}}=2{\pi }^2{f}^2\rho sv{A}^2$
where $\rho =$ density of medium
$s=$ cross sectional area of string

Given: $s^{\prime }=2s$
$\Rightarrow P_{\text{avg}}^{\prime }=2{\pi }^2{f}^2\rho \left(2s\right)v{A}^2$
$\therefore P_{\text{avg}}^{\prime }=2P_{\text{avg}}$
$\Rightarrow$Option $\left(a\right)$ is correct.