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 $v=\sqrt{\frac{V}{\rho n^{1/2}}}$