A mass $m$ moving horizontally with velocity $v_0$ strikes a pendulum of mass $m$. If the two masses stick together after the collision, then the maximum height reached by the pendulum is

A small block of mass $m$ slides along a smooth frictional track as shown in the figure. If it starts from rest at $P$ , velocity of block at point $Q$ is

A block of mass $1\,kg$ is pushed up a surface inclined to horizontal at an angle of $30^o$ by a force of $10\,N$ parallel to the inclined surface (figure). The coefficient of friction between block and the incline is $0.1$. If the block is pushed up by $10\,m$  along the inclined calculate 

$(a)$ work done against gravity

$(b)$ work done against force of friction

$(c)$ increases in potential energy

$(d)$ increases in kinetic energy

$(e)$ work done by applied force

$A$ particle of mass $m$ is released from $a$ height $H$ on $a$ smooth curved surface which ends into a vertical loop of radius $R$, as shown The minimum value of $H$ required so that the particle makes a complete vertical circle is given by

The kinetic energy $K$ of a particle moving along a circle of radius $R$ depends upon the distance $s$ as $K = as^2$. The force acting on the particle is

$ABCDE$ is a channel in the vertical plane, part $BCDE$ being circular with radius $r$ . A block is released from $A$ and slides without friction and without rolling. The block will complete the loop if $h$ is