A particle is moving along a circular path with a uniform speed. Through what angle does its angular velocity change when it completes half of the circular path?

A vehicle is moving with a velocity $v$ on a curved road of width $b$ and radius of curvature $R$. For counteracting the centrifugal force on the vehicle, the difference in elevation required in between the outer and inner edges of the rod is

A man revolves a stone of mass $m$ tied to the end of a string in a vertical circle of radius $R$. The net forces at the lowest and highest points of the circle directed vertical downwards are: [Here $T_1, T_2$ (and $v_1, v_2$) denote the tension in the string (and the speed of the stone) at the lowest and highest points, respectively.]

A small ring $P$ is threaded on a smooth wire bent in the form of a circle of radius $a$ and center $O$. The wire is rotating with constant angular speed $\omega$ about a vertical diameter $XY$, while the ring remains at rest relative to the wire at a distance $\frac{a}{2}$ from $XY$. Then ${\omega }^2$ is equal to