Question

Length of a string of density $\rho$ and Young's modulus $Y$ under tension is increased by $1 / n$ times of its original length. If the velocity of transverse and longitudinal vibrations of the string is same, find the value of such velocity.

• A. $\sqrt{\frac{Y}{\rho n}}$
• B. $\sqrt{\frac{Y}{\rho n^{1 / 2}}}$
• C. $\sqrt{\frac{Y}{\rho}}$
• D. $\sqrt{\frac{Y}{\rho n^{3 / 2}}}$

Answer 1

Answer: (B)

$v_{1}=\sqrt{\frac{Y}{\rho}}=$ velocity of longitudinal waves
$v_{2}=\sqrt{\frac{T}{\rho S}}=$ velocity of transverse waves
$v_{1} v_{2}=\frac{1}{\rho} \cdot \sqrt{Y . Y \frac{\Delta l}{l}}=\frac{Y}{\rho} \sqrt{\frac{\Delta l}{l}}$
$\left( \text{As} \frac{T}{S}=Y \frac{\Delta l}{l}\right)$
But $\frac{\Delta l}{l}=\frac{l / n}{l}=\frac{1}{n}$,
so, $v_{1} v_{2}=\frac{Y}{\rho n^{1 / 2}}$
or $ v^{2}=\frac{Y}{\rho n^{1 / 2}}$ or $v=\sqrt{\frac{V}{\rho n^{1 / 2}}}$