$A$ small block of mass $m$ is kept on a rough inclined surface of inclination $\theta$ fixed in an elevator. The elevator goes up with a uniform velocity $v$ and the block does not slide on the wedge. The work done by the force of friction on the block in time $t$ as seen by the observer on the inclined plane will be

$A$ particle of mass $m$ is moving along the $x$-axis with initial velocity $u \hat{i}$. It collides elastically with a particle of mass $10m$ at rest and then moves with half its initial kinetic energy (see figure). If $\sin \theta_{1} = \sqrt{n} \sin \theta_{2}$,then the value of $n$ is:

$A$ small bob $A$ of mass $m$ is attached to a massless rigid rod of length $1 \ m$ pivoted at point $P$ and kept at an angle of $60^{\circ}$ with the vertical as shown in the figure. At a distance of $1 \ m$ below point $P$, an identical bob $B$ is kept at rest on a smooth horizontal surface that extends to a circular track of radius $R$ as shown in the figure. If bob $B$ just manages to complete the circular path of radius $R$ up to a point $Q$ after being hit elastically by bob $A$, then the radius $R$ is . . . . . . $m$.

$A$ machine which is $75\%$ efficient, uses $12 \, J$ of energy in lifting up a $1 \, kg$ mass through a certain distance. The mass is then allowed to fall through that distance. The velocity at the end of its fall is (in $m \, s^{-1}$):

Three objects $A, B$ and $C$ are kept in a straight line on a frictionless horizontal surface. The masses of $A, B$ and $C$ are $m, 2m$ and $2m$ respectively. $A$ moves towards $B$ with a speed of $9 \ m/s$ and makes an elastic collision with it. Thereafter,$B$ makes a completely inelastic collision with $C$. All motions occur along the same straight line. The final speed of $C$ is $.... \ m/s$.

By how much should the spring be compressed by the block so that the centripetal force at point $P$ is $mg$?