Thermal Expansion

The temperature of an isotropic cubical solid of length $l_0$, density ${\rho }_0$ and coefficient of linear expansion $\alpha$ is increased by $20°\mathrm{C}$. Then at higher temperature, to a good approximation:-

A cylindrical metal rod of length $L_0$ is shaped into a ring with a small gap as shown. On heating the system
 

A wooden wheel of radius $R$ is made of two semicircular parts (see figure). The two parts are held together by a ring made of a metal strip of cross sectional area $S$ and length $L$. $L$ is slightly less than $2\pi R$. To fit the ring on the wheel, it is heated so that its temperature rises by $\mathrm{\Delta }T$ and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$, and its Young's modulus is $Y$, the force that one part of the wheel applies on the other part is

This question contains Statement 1 and Statement 2 Of the four choices given after the statements, choose the one that best describes the two statements.

Statement 1: The temperature dependence of resistance is usually given as $R=R_0\left(1+\alpha \mathrm{\Delta }t\right)$. The resistance of a wire changes from $100 \mathrm{\Omega }$ to $150 \mathrm{\Omega }$ when its temperature is increased from $27°\mathrm{C}$ to $227°\mathrm{C}$. This implies that $\alpha =2.5\times {10}^{-3}/{}^{\circ }\mathrm{C}$.

Statement 2: $R=R(1+\alpha \mathrm{\Delta }T)$ is valid only when the change in the temperature $\mathrm{\Delta }T$ is small and $\mathrm{\Delta }R=\left(R-R_0\right)<<R_0$.

$2 \mathrm{kg}$ 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)

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 ${\alpha }_1:{\alpha }_2$.

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 ${10}^{-5}/{}^{\circ }\mathrm{C}$)

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 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?

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

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 bolt is passed through a pipe and a nut is just tightened. Coefficients of linear expansion for bolt and pipe material are ${\alpha }_b$ and ${\alpha }_p$ respectively. If the assembly is heated then:

Two metal plates $A$ and $B$ made of same material are placed on a table as shown in the figure. If the plates are heated uniformly, the gaps indicated by $x$ and $y$ in the figure,

A vessel is partly filled with liquid. When the vessel is cooled to a lower temperature, the space in the vessel unoccupied by the liquid remains constant. Then the volume of the liquid $\left(V_L\right)$ volume of the vessel $\left(V_v\right)$, the coefficient of cubical expansion of the material of the vessel $\left({\gamma }_V\right)$ and of the solid $\left({\gamma }_L\right)$ are related as

A thin cylindrical metal rod is bent into a ring with a small gap as shown in figure. On heating the system,

A platinum sphere floats in mercury. Find the percentage change in the fraction of volume of sphere immersed in mercury when the temperature is raised by $80°\mathrm{C}$ (volume expansivity of mercury is $182\times {10}^{-6}/{}^{\circ }\mathrm{C}$ and linear expansivity of platinum is $9\times {10}^{-6}/{}^{\circ }\mathrm{C}$).

The apparent coefficient of expansion of a liquid when heated, filled in vessel $A$ and $B$ of identical volumes, is found to be ${\gamma }_1$ and ${\gamma }_2$, respectively. If ${\alpha }_1$ be the linear expansivity of $A$ then that of $B$ will be

The densities of wood and benzene at $0{}^{\circ }\mathrm{C}$ are $880 \mathrm{kg} {\mathrm{m}}^{-3}$ and $900 \mathrm{kg} {\mathrm{m}}^{-3}$, respectively, The coefficients of volume expansion of wood is $1.2\times {10}^{-3}/{}^{\circ }\mathrm{C}$ and of benzene $1.5\times {10}^{-3}/{}^{\circ }\mathrm{C}$. The temperature at which a piece of this wood would just sink in benzene at the same temperature is

A brass wire $2 \mathrm{m}$ long at $27°\mathrm{C}$ is held taut with negligible tension between two rigid supports. If the wire is cooled to a temperature of $-33°\mathrm{C}$, then the tension developed in the wire, its diameter being $2 \mathrm{mm}$, will be (coefficient of linear expansion of brass $=2.0\times {10}^{-5}/{}^{\circ }\mathrm{C}$ and Young's modulus of brass $=0.91\times {10}^{11} \mathrm{Pa}$)

Two elastic rods are joined between fixed supports as shown in the figure. Condition for no change in the lengths of individual rods with the increase of temperature (${\alpha }_1,{\alpha }_2=$ linear expansion coefficient, $A_1,A_2=$ area of rods, $Y_1,Y_2=$ Young's modulus) is

The coefficient of volumetric expansion of mercury is $18\times {10}^{-5}/{}^{\circ }\mathrm{C}$. A thermometer bulb has a volume ${10}^{-6}{ \mathrm{m}}^3$ and cross section of stem is of $0.004{ \mathrm{cm}}^2$. Assuming that bulb is filled with mercury at $0°\mathrm{C}$, the length of the mercury column at $100°\mathrm{C}$ is