A wire of length $l$ having tension $T$ and radius $r$ vibrates with natural frequency $f$. Another wire of…
A wire of length $l$ having tension $T$ and radius $r$ vibrates with natural frequency $f$. Another wire of same metal with length $2l$ having tension $2T$ and radius $2r$ will vibrate with natural frequency
$f$
$2f$
$2\sqrt{2} f$
$\frac{f}{2\sqrt{2}}$
Solution
$f = \frac{v}{2l} = \frac{\sqrt{T/\alpha}}{2l} = \frac{\sqrt{T/\rho s}}{2l} = \frac{\sqrt{T/\pi r^2 \rho}}{2l} \quad (\because s = \pi r^2)$
$\therefore f \propto \frac{\sqrt{T}}{rl}$
Now, tension, length and radius all are doubled.
Hence, $f' = \frac{f}{2\sqrt{2}}$