Question

A small body of mass $m$ tied to a non-stretchable thread moves over a smooth horizontal plane. The other end of the thread is being drawn into a hole $O$ (figure shown above) with a constant velocity. Find the thread tension as a function of the distance $r$ between the body and the hole if at $r=r_0$ the angular velocity of the thread is equal to ${\omega }_0$.

Answer 1

Forces, acting on the mass $m$ are shown in figure below. As $\overrightarrow{N}=m\overrightarrow{g}$, the net torque of these two forces about any fixed point must be equal to zero. Tension $T$, acting on the mass $m$ is a central force, which is always directed towards the centre $O$. Hence the moment of force $T$ is also zero about the point $O$ and therefore the angular momentum of the particle $m$ is conserved about $O$.
Let, the angular velocity of the particle be $\omega$, when the separation between hole and particle $m$ is $r$, then from the conservation of momentum about the point $O$,
$m\left({\omega }_0{r}_0\right)r_0=m\left(\omega r\right)r$,
or $\omega =\frac{{\omega }_0{r}_0^2}{r^2}$
Now, from the second law of motion for $m$,
$T=F=m{\omega }^{2}r$
Hence the sought tension,
$F=\frac{m{\omega }_0^2{r}_0^{4}r}{r^4}=\frac{m{\omega }_0^2{r}_0^4}{r^3}$