$A$ light inextensible string that goes over a smooth fixed pulley as shown in the figure connects two blocks of masses $m_1 = 0.36 \text{ kg}$ and $m_2 = 0.72 \text{ kg}$. Taking $g = 10 \text{ m/s}^2$,find the work done (in joules) by the string on the block of mass $0.36 \text{ kg}$ during the first second after the system is released from rest.

Two rectangular blocks $A$ and $B$ of masses $2 \,kg$ and $3 \,kg$ respectively are connected by a spring of spring constant $10.8 \,Nm^{-1}$ and are placed on a frictionless horizontal surface. The block $A$ is given an initial velocity of $0.15 \,ms^{-1}$ in the direction shown in the figure. The maximum compression of the spring during the motion is (in $\,m$)

$A$ box of mass $1 \ kg$ is placed on a horizontal surface of length $1 \ m$. $A$ force of $8 \ N$ is applied to move it horizontally by $1 \ m$,while simultaneously,its vertical height is increased by $2 \ m$. What is the total work done in Joules? (Take $g = 10 \ m/s^2$)

In a perfectly inelastic collision, two spheres made of the same material with masses $15 \ kg$ and $25 \ kg$, moving in opposite directions with speeds of $10 \ m/s$ and $30 \ m/s$, respectively, strike each other and stick together. The rise in temperature (in $^{\circ}C$), if all the heat produced during the collision is retained by these spheres, is: (specific heat of sphere material $31 \ cal/kg \cdot ^{\circ}C$ and $1 \ cal = 4.2 \ J$)

$A$ curved surface is shown in the figure. The portion $BCD$ is free of friction. There are three spherical balls of identical radii and masses. Balls are released from rest one by one from $A$,which is at a slightly greater height than $C$.
With the surface $AB$,ball $1$ has large enough friction to cause rolling down without slipping; ball $2$ has a small friction and ball $3$ has a negligible friction.
$(a)$ For which balls is total mechanical energy conserved?
$(b)$ Which ball$(s)$ can reach $D$?
$(c)$ For balls which do not reach $D$,which of the balls can reach back $A$?

Two bodies $A$ and $B$ of masses $2m$ and $m$ are projected vertically upwards from the ground with velocities $u$ and $2u$ respectively. The ratio of the kinetic energy of body $A$ and the potential energy of body $B$ at a height equal to half of the maximum height reached by body $A$ is (in $ : 1$)