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) 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`.

Explanation

Forces, acting on the mass m are shown in the figure. As `vecN=mvecg`, 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(omega_0r_0)r_0=m(omegar)r`,
or `omega=(omega_0r_0^2)/(r^2)`
Now, from the second law of motion for m,
`T=F=momega^2r`
Hence the sought tension,
`F=(momega_0^2r_0^4r)/(r^4)=(momega_0^2r_0^4)/(r^3)`