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A mass m moves in a circle on a smooth horizontal plane with velocity $v_0$ 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
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
A
$mv_0^2$
B
$\frac{1}{4}mv_0^2$
C
$2mv_0^2$
D
$\frac{1}{2}mv_0^2$
Detailed Solution
The tension is a central force, so angular momentum about the hole is conserved.
$mv_0R_0 = mv\frac{R_0}{2} \Rightarrow v = 2v_0$
$KE_f = \frac{1}{2}m(2v_0)^2 = 2mv_0^2$
$mv_0R_0 = mv\frac{R_0}{2} \Rightarrow v = 2v_0$
$KE_f = \frac{1}{2}m(2v_0)^2 = 2mv_0^2$