The heat energy required to convert $10 \,kg$ of ice at $-10^{\circ} C$ into water at $0^{\circ} C$ is (specific heat capacity of ice $=0.5 \,cal \,g^{-1} \,^{\circ}C^{-1}$ and latent heat of fusion of ice $=80 \,cal \,g^{-1}$)

An iron rod of heat capacity $C$ is heated to temperature $8T_0$. It is then put in a cylindrical vessel of adiabatic walls having two moles of air which can be treated as a diatomic ideal gas at temperature $T_0$ and closed by a movable piston which is also adiabatic. The atmospheric pressure is $P_0$. The cylinder with the piston combined have a heat capacity $2C$. Find the equilibrium temperature. (Assume the temperature of air to be uniform and equal to the vessel at all times).

$A$ beaker is filled with water at $4\,^{\circ}C$. At one time the temperature is increased by a few degrees above $4\,^{\circ}C$ and at another time it is decreased by a few degrees below $4\,^{\circ}C$. One shall observe that:

$A$ piece of ice falls from a height $h$ so that it melts completely. Only one-quarter of the heat produced is absorbed by the ice and all energy of ice gets converted into heat during its fall. The value of $h$ is (Latent heat of ice is $L = 3.4 \times 10^{5} \text{ J/kg}$ and $g = 10 \text{ N/kg}$) (in $\text{ km}$)

$A$ lead bullet at $27\ ^oC$ just melts when stopped by an obstacle. Assuming that $25\%$ of heat is absorbed by the obstacle,the velocity of the bullet at the time of striking is ....... $m/s$. (Melting point of lead $= 327\ ^oC$,specific heat of lead $= 0.03\ cal/g\ ^oC$,latent heat of fusion of lead $= 6\ cal/g$,and $J = 4.2\ J/cal$)

$A$ $10.0 \,W$ electrical heater is used to heat a container filled with $0.5 \,kg$ of water. It is found that the temperature of the water and the container rose by $3 \,K$ in $15 \,min$. The container is then emptied,dried,and filled with $2 \,kg$ of an oil. It is now observed that the same heater raises the temperature of the container-oil system by $2 \,K$ in $20 \,min$. Assuming no other heat losses in any of the processes,the specific heat capacity of the oil is ................ $\times 10^3 \,J K^{-1} kg^{-1}$.