A circular turntable has a block of ice placed at its centre. The system rotates with an angular speed $\text{ω}$ about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system,

A thin ring of mass $2 m$ and radius $R$ is pivoted at $P$ on a frictionless table, as shown in figure. A bug of mass $m\mathit{.}$ runs along the ring with speed $v$ with respect to the ring. The bug starts from the pivot with the ring at rest. When the bug reaches the diametrically opposite point on the ring (point $X$), it is moving with velocity $\text{'}v\text{'}$ with respect to the table. The value of $\frac{v}{v^{\text{'}}}$ is

A particle starts from the point $(0 \mathrm{m},8\mathrm{m})$ and moves with uniform velocity of $3\hat{\mathrm{i}}{\mathrm{ms}}^{-1}.$ After $5$ seconds, the angular velocity of the particle about the origin will be