Figure shown the roller coaster track. Each car will start from rest at point $A$ and will roll with negligible friction. If is important that there should be at least some small positive normal force exerted by the track on the car at all points, otherwise the car would leave the track. With the above fact, the minimum safe value for the radius of curvature at point $B$ is $\left(g=10{\mathrm{ms}}^2\right):$

 

A ring rotates about $z$ axis as shown in figure. The plane of rotation is $xy$. At a certain instant the acceleration of a particle $P$ (shown in figure) on the ring is $\left(6\hat{i}-8\hat{j}\right) \mathrm{m} {\mathrm{s}}^{-2}$. Find the magnitude of angular acceleration in $\mathrm{rad} {\mathrm{s}}^{-2}$. Radius of the ring is $2 \mathrm{m}$.

Figure shows the total acceleration and velocity of a particle moving clockwise in a circle of radius $2.5 \mathrm{m}$ at a given instant of time. At this instant. Find the radial acceleration in $\mathrm{m} {\mathrm{s}}^{-2}$.

A mass $m$ rotating freely in a horizontal circle of radius $1\mathrm{m}$ on a frictionless smooth table supports a stationary mass $2\mathrm{m}$, attached to the other end of the string passing through smooth hole $O$ in table, hanging vertically. Find the angular velocity of rotation.

A circular platform rotates around a vertical axis with angular velocity $\omega =10 {\mathrm{rads}}^{-1}$. On the platform is a ball of mass $1\mathrm{kg}$, attached to the long axis of the platform by a thin rod of length $10\mathrm{cm}$ $\left(\alpha =30°\right)$. Find normal force exerted by the ball on the platform (in newton). Friction is absent.