A truck moves with a uniform velocity $v$ through a convex bridge and a concave bridge having the same radius $R$. Show that the ratio of the forces exerted by truck at the highest point of the convex bridge and the lowest point of the concave bridge is ($gR-v^2$) : ($gR+v^2$), $g$ being the acceleration due to gravity.

A particle rotates about a vertical axis in a horizontal circle according to the law $\theta =at-b{t}^3$, where $a=9 \mathrm{rad}/\mathrm{s}$ and $b=3 \mathrm{rad}/{\mathrm{s}}^3$. Find the mean values of the angular velocity and acceleration over the time interval between $t=0$ and time when the particle stops.

A cylindrical drum, diameter $20 \mathrm{cm}$, rotates about its axis which is vertical. A small steel body in contact with the upper part of the inner wall is carried round with the drum if the rate of rotation exceeds $200$ revolution per minute, but falls at lower rates. Find the coefficient of limiting friction between the steel surfaces.

There is a big sphere having smooth outside surface. The radius of the sphere is $r$. A particle of $m$ is situated at the top of the sphere. What velocity may be given to it in the horizontal direction such that the particle slides down the surface of the sphere and leaves the surface when it is at a vertical height of $\frac{3}{4}r$ from the centre of the sphere?

A pellet of mass $1 \mathrm{g}$ is moving with an angular velocity of $1 \mathrm{rad}/\mathrm{s}$ along a circle of $1$ metre. The centrifugal force is

A small coin is placed on a whirling table. The coin just begins to slip when it is placed $4 \mathrm{cm}$ from the centre of the table. If the table rotates with double the angular speed, at what distance from the centre of the table should the coin be placed so that it just starts slipping?