$A$ uniform flexible chain of mass $m$ and length $2l$ hangs in equilibrium over a smooth horizontal pin of negligible diameter. What will be the speed of the chain when it leaves the pin?

$A$ particle of mass $m$ is moving along a circle of radius $R$ such that its tangential acceleration $a_t$ varies with distance covered $x$ as $a_t = \alpha x^2$, where $\alpha$ is a constant. The kinetic energy $K$ of the particle varies with the distance as $K = \beta x^c$, where $\beta$ and $c$ are constants. The values of $\beta$ and $c$ are:

$A$ ball of mass $1 \ kg$ moves in a straight line with velocity $v = c x^\alpha$, where $c = 1$ ($SI$ unit) and $\alpha$ is a constant. If the work done by the net force during its displacement from $x = 0$ to $x = 4 \ m$ is $128 \ J$, then the value of $\alpha$ is

$A$ body of mass $m$ slides down a smooth hemispherical surface of radius $r$. What will be its velocity at the bottom of the surface?

The average force necessary to stop a bullet of mass $20\, g$ moving with a speed of $250\, m/s$,as it penetrates into the wood for a distance of $12\, cm$ is

$A$ body of mass $M$ is dropped from a height $h$ on a sand floor. If the body penetrates $x$ into the sand,the average resistance offered by the sand to the body is