Question

Length of a string of density ρ and Young’s modulus Y under tension is increased by $\frac{1}{n}$ 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

The correct option is B $\sqrt{\frac{Y}{\rho n^{1/2}}}$

$V_1=\sqrt{\frac{Y}{\rho }}=\text{velocity of longitudinal waves}{V}_2=\sqrt{\frac{T}{\rho A}}=\text{velocity of transverse waves}{V}_1{V}_2=\frac{1}{\rho }.\sqrt{Y.Y\frac{\Delta l}{l}}=\frac{Y}{\rho }\sqrt{\frac{\Delta l}{l}}(As\frac{T}{A}=Y\frac{\Delta l}{l})But\frac{\Delta l}{l}=\frac{l/n}{l}=\frac{1}{n}{V}_1{V}_2=\frac{Y}{\rho n^{1/2}}V=\sqrt{\frac{Y}{\rho n^{1/2}}}$