Question

A small body $A$ is fixed to the inner side of a thin rigid hoop of radius $R$ and mass equal to that of the body $A$. The hoop rolls without slipping over a horizontal plane. At the moments when the body $A$ gets into the lower position, the centre of the hoop moves with velocity $v_o$. At what values of $v_o$ will the hoop move without bouncing?

Answer 1

The hoop jumps if centrifugal force on A $>$ weight of hoop $+$ A.

For pure rotation, velocity of A at topmost point$=2v_0^{{}^{\prime }}$

$\Rightarrow m\frac{(2v_0^{{}^{\prime }}{)}^2}{R}=4mg$

$\Rightarrow v_0^{{}^{\prime }}=4Rg$

Now, Conservation of Energy gives

$\frac{1}{2}I({\omega }_0^{{}^{\prime }}{)}^2+\frac{1}{2}m(v_0^1{)}^2+\frac{1}{2}m(2v_0^1{)}^2+2R\cdot mg=\frac{1}{2}I{\omega }_0^2+\frac{1}{2}m{v}_0^2+\frac{1}{2}m(v_0^2)$

$\Rightarrow 2\times 12Rg=3v_0^2$

$v_0^2=8Rg$ or $v_0=2\sqrt{2Rg}$ or $\sqrt{8Rg}=v_0$