Question

A rod of length $l=2m$ is maintained to rotate with a constant angular velocity $\omega =1rad{s}^{-1}$ about a vertical axis passing through one end (diagram). There is a spring constant $k=1N{m}^{-1}$ which just encloses rod inside it in natural length. One end of the spring is attached to the axis of rotation. $S$ is a sleeve of mass $m=1kg$ which can just fit on rod. All surfaces are smooth. With what minimum kinetic energy (in $j$) sleeve should be projected so that it enters on the rod without jerk and completely compresses the spring.

Answer 1

Answer: 6

For entering without jerk $v_2=l{\omega }_0=2m{s}^{-1}$
Using work-energy theorem on the sleeve after entering in the frame of rod

$W_{\text{spring }}+W_{\text{centrifugal }}=\Delta K$
$-\frac{1}{2}k{l}^2-\frac{1}{2}m{\omega }^2{l}^2=0-\frac{1}{2}m{v}_1^2$
$\Rightarrow v_1^2=8$
Now, $K=\frac{1}{2}m\left(v_1^2+v_2^2\right)=6$