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. Find the new value of speed of the wave.

• A. $\sqrt{\frac{Y}{pn}}$
• B. $(\frac{Y}{pn}{)}^{\frac{-1}{2}}$
• C. $\sqrt{\frac{Y}{p}}$
• D. $(\frac{Y}{pn}{)}^{\frac{3}{2}}$

Answer 1

The correct option is A $\sqrt{\frac{Y}{pn}}$

For longitudnal vibrations, $v=\sqrt{\frac{Y}{p}}$,
$Y=\frac{stress}{strain}=\frac{stress}{\left(1/n\right)}$
$stress=\frac{Tension}{\text{area of cross section of wire}A}=\frac{Y}{n}$
$Tension=\frac{YA}{n}$
For transverse vibrations, $v=\sqrt{\frac{Tension}{\text{mass of the string of per unit length}}}$
$v=\sqrt{\frac{\frac{YA}{n}}{\text{mass of the string of per unit length}}}$
$v=\sqrt{\frac{\frac{YA}{n}}{pA}}=\sqrt{\frac{Y}{np}}$
Option "A" is correct.