$2 \mathrm{kg}$ of ice at $-20°\mathrm{C}$ is mixed with $5 \mathrm{kg}$ of water at $20°\mathrm{C}$ in an insulating vessel having a negligible heat capacity. Calculate the final mass of water (in $\mathrm{kg}$) remaining in the container.

$2 \mathrm{kg}$ of ice at $-15°\mathrm{C}$ is mixed with $2.5 \mathrm{kg}$ of water at $25°\mathrm{C}$ in an insulating container. If the specific heat capacities of ice and water are $0.5 \mathrm{cal}/\mathrm{g}°\mathrm{C}$ and $1 \mathrm{cal}/\mathrm{g}°\mathrm{C}$, find the amount of water present in the container? (in $\mathrm{kg}$ nearest integer)

In two experiments with a continuous flow calorimeter to determine the specific heat capacity of a liquid, an input power of $16 \mathrm{W}$ produced a rise of $10 \mathrm{K}$ in the liquid. When the power was doubled, the same temperature rise was achieved by making the rate of flow of liquid three times faster. Find the power lost (in $\mathrm{W}$) to the surrounding in each case.

A clock with a metallic pendulum at $15°\mathrm{C}$ runs faster by $5 \mathrm{s}$ each day and at $30°\mathrm{C}$, runs slow by $10 \mathrm{s}$. Find the coefficient of linear expansion of the metal. (nearly in ${10}^{-5}/{}^{\circ }\mathrm{C}$)

Two vessels connected at the bottom by a thin pipe with a sliding plug contain liquid at $20°\mathrm{C}$ and $80°\mathrm{C}$ respectively. The coefficient of cubic expansion of liquid is ${10}^{-3}{ \mathrm{K}}^{-1}$. The ratio of heights of liquid columns in the vessel $\left(h_{20}/h_{80}\right)$ is nearest to which integer?

Three rods of equal length $l$ are joined to form an equilateral triangle $PQR$. $O$ is the mid point of $PQ$. Distance $OR$ remains the same for small change in temperature. Coefficient of linear expansion $PR$ and $RQ$ is the same, i.e., ${\alpha }_2$ but that for $PQ$ is ${\alpha }_1$. Then find the ratio ${\alpha }_1:{\alpha }_2$.

We would like to increase the length of $15 \mathrm{c}\mathrm{m}$ long copper rod of cross section $4 \mathrm{m}{\mathrm{m}}^2$ by $1 \mathrm{m}\mathrm{m}$. The energy absorbed by the rod if it is heated is $E_{\mathit{1}}$. The energy absorbed by the rod if it is stretched slowly is $E_{\mathit{2}}$. Then, find the value of  $\frac{E_1}{10E_2}$. [Various parameters of copper are density$=9\times {10}^3 \mathrm{k}\mathrm{g} {\mathrm{m}}^{-3}$, thermal coefficient of linear expansion$=16\times {10}^{-6} {\mathrm{K}}^{-1}$, Young’s modulus$=135\times {10}^9$ $\mathrm{Pa}$, specific heat$=400 \mathrm{J} \mathrm{k}{\mathrm{g}}^{-1} {\mathrm{K}}^{-1}$]

A rod has variable co-efficient of linear expansion $\alpha =\frac{x}{5000}$. If length of the rod is $1 \mathrm{m}$. Determine increase in length of the rod in $\left(\mathrm{cm}\right)$ on increasing temperature of the rod by $100°\mathrm{C}$