A stone of mass $2 \mathrm{~kg}$ is tied at one end of a string of length $2 \mathrm{~m}$ and is whirled in…
A stone of mass $2 \mathrm{~kg}$ is tied at one end of a string of length $2 \mathrm{~m}$ and is whirled in a horizontal circle. If the string can withstand a maximum tension of $64 \mathrm{~N}$, then the permissible maximum number of rotations per minute is
19
$\frac{60}{\pi}$
$\frac{152}{3} \pi$
$\frac{120}{\pi}$
Solution
Mass of stone, $\mathrm{m}=2 \mathrm{~kg}$
Length of the string $=$ radius, $r=2 \mathrm{~m}$
Maximum tension, $T_{\max }=64 \mathrm{~N}$
$\begin{aligned} & \mathrm{T}_{\max }=\frac{\mathrm{mv}_{\max }^2}{\mathrm{r}} \\ & 64=\frac{2 \times \mathrm{v}_{\max }^2}{2} \\ & \mathrm{v}_{\max }=8 \mathrm{~m} / \mathrm{s} \\ & \mathrm{v}_{\max }=\mathrm{r} \omega_{\max } \\ & 8=2 \times 2 \pi \mathrm{f} \\ & \mathrm{f}=\frac{2}{\pi} \times 60=\frac{120}{\pi}\end{aligned}$
Hence, the permissible maximum number of rotations per minute is $\frac{120}{\pi}$.