A small ball of mass $1 \mathrm{kg}$ and charge $8\mu C$ is placed at the centre of a uniformly charged non-conducting solid sphere of radius $2 \mathrm{m}$ and charge $\frac{1}{6} \mathrm{mC}$. A narrow smooth groove is made in the sphere from centre to the surface of sphere and the sphere is made to rotate about its vertical diameter at a constant rate of $\frac{1}{4\pi }$ revolution per second. Find the speed $\left(\mathrm{in} \mathrm{m} {\mathrm{s}}^{-1}\right)$ with respect to ground with which the ball slides out from the groove. Ignore any magnetic force acting on the ball. (Take $\sqrt{2}=1.41$)

The Earth (of mass $6\times {10}^{24} \mathrm{kg}$ ) revolves round the sun with an angular velocity of $2\times {10}^{-7} \mathrm{rad} {\mathrm{s}}^{-1}$ in a circular orbit of radius $1.5\times {10}^{11} \mathrm{km}.$ The force exerted by the sun on the earth is $\underline{ }$
 

Two spheres of equal masses are attached to a string of length $2 \mathrm{m}$ as shown in figure. The string and the sphere are then whirled in a horizontal circle about $O$ at a constant rate. What is the value of the ratio?
$\frac{\left(\mathrm{Tension} \mathrm{in} \mathrm{the} \mathrm{string} \mathrm{between} P \mathrm{and} Q\right)}{\left(\mathrm{Tension} \mathrm{in} \mathrm{the} \mathrm{string} \mathrm{between} P \mathrm{and} O\right)}=$?

Two stones of masses $m$ and $2m$ are whirled in horizontal circles, the heavier one in radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of lighter stone is $n$ times that of the tangential speed of the heavier stone. They are reported to experience the same centripetal force. The value of $n$ is, 

A smooth circular groove has a smooth vertical wall as shown in figure. A block of mass $m$ moves against the wall with a speed $v$. Which of the following curve represents the correct relation between the normal reaction on the block by the wall $\left(N\right)$ and speed of the block $\left(v\right)$?