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.

One end of a massless spring of spring constant $100 \mathrm{N} {\mathrm{m}}^{-1}$ and of natural length $50 \mathrm{cm}$ is fixed and the other end is connected to a mass of $0.5 \mathrm{kg}$ lying on a frictionless horizontal table. The spring remains unstretched on the table. If the mass now rotates with an angular velocity $2 \mathrm{rad} {\mathrm{s}}^{-1}$, find the elongation of the spring.

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 small body of mass $m$ rests at the top of a smooth hemisphere of radius $r$. It is displaced so that it slides over the hemisphere. Show that the body will lose contact with the hemisphere when it descends vertically through a height $\frac{r}{3}$ from the top.

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.

A man whirls a stone round his head on the end of string $1 \mathrm{m}$ long. Can the string be in a horizontal plane? If the stone has a mass $200 \mathrm{gm}$ and the string will snap if the tension in it exceeds $4 \mathrm{N}$, what is the smallest angle the string can make with the horizontal? What is then the speed of the stone? Take $g=10 \mathrm{m}/{\mathrm{s}}^2$.

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?