Each small part of the ring will experience a centrifugal force radially outwards. So the ring will tend to expand, i.e. the radius and circumference will tend to increase. By virtue of its elasticity the ring will oppose its expansion. So each part of the ring will experience a force of pull or tension from the other part.
Consider a small part ACB of the ring that subtends an angle ∆θ at the center as shown in the Figure. Let the tension in the ring be T.
The forces on this elementary portion ACB are:
(i) Tension T by the part of the ring left to A
(ii) Tension T by the part of the ring right to B
(iii) Weight (∆m)g
(iv) Normal force N by the table
As the elementary portion ACB moves in a circle of radius R at constant speed v, its acceleration toward the centre is \(\frac{(\Delta m)v^2}R\). Resolving the forces along the radius CO
Thus the length of the part ACB = R ∆θ . The mass per unit length of the ring is \(\frac m{2\pi R}\).
∴ Mass of this portion ACB,
Putting the value of ∆m in (ii)
