Question

Two vibrating strings of the same material but length $L$ and $2L$ have radii $2r$ and $r$ respectively. They are stretched under the same tension. Both the string vibrate in their fundamental modes, the one of length $L$ with frequency $v_1$ and other with frequency $v_2$. The ratio $v_1/v_2$ is given by

• A. $2$
• B. $4$
• C. $8$
• D. $1$

Answer 1

The correct option is D $1$

For vibrating string

frequency $\alpha \frac{1}{lenght}\sqrt{\frac{Tension}{massperunitlenght}}$ ---- $\left(1\right)$

Since area of first wire is $\frac{\pi (2{)}^2}{\pi 1^2}=4$ times pf second wire

its mass per unit length is also $4$ times

i.e. ${\mu }_1=4\mu$ ${\mu }_2=\mu$

$v_1=\frac{1}{L}\sqrt{\frac{T}{4\mu }}=\frac{1}{\mu }\sqrt{\frac{T}{\mu }}$

$v_2=\frac{1}{2}\sqrt{\frac{T}{\mu }}$

$\frac{v_1}{v_2}=\frac{1}{2L}\sqrt{\frac{T}{\mu }}\times \frac{2l}{1}\sqrt{\frac{\mu }{T}}$

$=1$