First slide

Force

Question

A mass m moves in a circle on a smooth horizontal plane with velocity $ν_{0}$ at a radius $R_{0}$ . The mass is attached to a string which passes through a smooth hole in the plane as shown.
The tension in the string is increased gradually and finally m moves in a circle of radius $\frac{R_{0}}{2}$ . The final value of the kinetic energy is

Moderate

A

$2 m v_{0}^{2}$
B

$\frac{1}{2} m v_{0}^{2}$
C

$m v_{0}^{2}$
D

$\frac{1}{4} m v_{0}^{2}$

Solution

According to law of conservation of angular momentum
$m v r = m v^{'} r^{'}$ . . . .(i)
$v_{0} R_{0} = v (\frac{R_{0}}{2}); v = 2 v_{0}$
$∴ \frac{K_{0}}{K} = \frac{\frac{1}{2} m v_{0}^{2}}{\frac{1}{2} m v^{2}} = {(\frac{v_{0}}{v})}^{2}$
$or \begin{array}{l} \frac{K}{K_{0}} = {(\frac{v}{v_{0}})}^{2} = (2)^{2} \\ K = 4 K_{0} = 2 m v_{0}^{2} \end{array}$ . . . .(using (i))

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