Question

A circular table with smooth horizontal surface is rotating at an angular speed $\omega$ about its axis. A groove is made on the surface along a radius, and a small particle is gently placed inside the groove at a distance l from the center. Find the speed of the particle with respect to the table as its distance from the center becomes L.

Answer 1

As no external force is acting on the system the angular momentum must be conserved about the axis of rotation.

So, $P=I{W}^{2}R$

$\frac{1}{2}{MR}^2{W}^{2}R+{ml}^2{w}^{2}l=\frac{1}{2}{MR}^2{W}^{2}R+{ML}^2{W}_f^{2}L$

$\Rightarrow {ml}^3{w}^2={mL}^3{W}_f^2$

$W_f={\left(\frac{l}{L}\right)}^{3/2}W$

Speed $={LW}_f=\frac{l^{3/2}}{L^{1/2}}W$