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 $100°\mathrm{C}$

A clock pendulum made of invar has a period of $2 \mathrm{s}$ at $20°\mathrm{C}$. If the clock is used in a climate where average temperature is $40°\mathrm{C}$, what correction (in seconds) may be necessary at the end of $10$ days to the time given by clock?

$\left({\alpha }_{\mathrm{invar}}=7\times {10}^{-7\circ }{\mathrm{C}}^{-1}, \mathrm{day}=8.64\times {10}^4 \mathrm{s}\right)$

The temperature of a body rises by $44°\mathrm{C}$ when a certain amount of heat is given to it. The same heat when supplied to $22 \mathrm{g}$ of ice at $-8°\mathrm{C}$, raises its temperature by $16°\mathrm{C}$. Find the water equivalent of the body in grams [Given: $s_{\mathrm{water}}=1 \mathrm{cal}/\mathrm{g}°\mathrm{C}$ and $L_f=80 \mathrm{cal}/\mathrm{g}, s_{\mathrm{ice}}=0.5 \mathrm{cal}/\mathrm{g}°\mathrm{C}$]

A fine steel wire of length $4 \mathrm{m}$ is fixed rigidly in a heavy brass frame as shown in figure. It is just taut at $20°\mathrm{C}$. Find the the tensile stress developed in steel wire($n$ $\times {10}^6 {\mathrm{Nm}}^{-2}$) if whole system is heated to $120°\mathrm{C}$.

(Given ${\alpha }_{\mathrm{brass}}=1.8\times {10}^{-5\circ }{\mathrm{C}}^{-1}, {\alpha }_{\mathrm{steel}}=1.2\times {10}^{-5\circ }{\mathrm{C}}^{-1}, Y_{\mathrm{steel}}=20\times {10}^{11}{\mathrm{Nm}}^{-2}, Y_{\mathrm{brass}}=1.7\times {10}^7{\mathrm{Nm}}^{-2}$)

A lead ball at $25°\mathrm{C}$ is dropped from a height of $2 \mathrm{km}$. It is heated due to air resistance and it is assumed that all of its kinetic energy is used in increasing the temperature of ball. If specific heat of lead is $s=125 \mathrm{Joule}/\mathrm{kg}{}^{\circ }\mathrm{C}$, find the final temperature of ball in ${}^{\circ }\mathrm{C}$.

A certain mass of a solid exists at its melting temperature of $20°\mathrm{C}$. When a heat $Q$ is added $3/4$ of the material melts. When an additional $Q$ amount of heat is added the material transforms to its liquid state at $50°\mathrm{C}$. Find the ratio of specific latent heat of fusion (in $\mathrm{J}/\mathrm{g}$) to the specific heat capacity of the liquid (in ${\mathrm{Jg}}^{-1}{{}^{\circ }\mathrm{C}}^{-1}$) for the material.