B M Sharma Solutions for Chapter: Thermometry, Thermal Expansion and Calorimetry, Exercise 4: Archives

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 $\mathrm{\Delta }R=\left(R-R_0\right)<<R_0$.

$100 \mathrm{g}$ of water is heated from $30 °\mathrm{C}$ to $50 °\mathrm{C}$. Ignoring the slight expansion of the water, the change in its internal energy is (Specific heat of water is $4184 \mathrm{J} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$)

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

The pressure that has to be applied to the ends of a steel wire of length $10 \mathrm{cm}$ to keep its length constant when its temperature is raised by $100°\mathrm{C}$ is (For steel, Young's modulus is $2\times {10}^{11} \mathrm{N} {\mathrm{m}}^{-2}$ and coefficient of thermal expansion is $1.1\times {10}^{-5} {\mathrm{K}}^{-1}$)

A pendulum clock lose $12 \mathrm{s}$ a day if the temperature is, $40°\mathrm{C}$ and gains, $4 \mathrm{s}$ a day if the temperature is, $20°\mathrm{C}.$ The temperature at which the clock will show correct time, and the co-efficient of linear expansion $\left(\alpha \right)$ of the metal of the pendulum shaft are respectively :

A copper ball of mass $100 \mathrm{gm}$ is at a temperature $T$. It is dropped in a copper calorimeter of mass $100 \mathrm{gm}$, filled with $170 \mathrm{gm}$ of water at room temperature. Subsequently, the temperature of the system is found to be $75 °\mathrm{C}$. $T$ is given by: (Given: room temperature $=30 °\mathrm{C}$, the specific heat of copper $=0.1 \mathrm{cal} {\mathrm{gm}}^{-1}°{\mathrm{C}}^{-1}$)

A water cooler of storage capacity $120 \mathrm{litres}$ can cool water at a constant rate of $P \mathrm{watts}$. In a closed circulation system (as shown schematically in the figure), the water from the cooler is used to cool an external device that generates constantly $3 \mathrm{kW}$ of heat (thermal load). The temperature of water fed into the device cannot exceed $30 °\mathrm{C}$ and the entire stored $120 \mathrm{litres}$ of water is initially cooled to $10 °\mathrm{C}$. The entire system is thermally insulated. The minimum value of $P$ (in $\mathrm{watts}$) for which the device can be operated for $3 \mathrm{hours}$ is

(specific heat of water is $4.2 \mathrm{kJ} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$ and the density of water is $1000 \mathrm{kg} {\mathrm{m}}^{-3}$).

The ends $Q$ and $R$ of two thin wires, $PQ$ and $RS,$ are soldered (joined) together. Initially, each of the wire has a length of $1 \mathrm{m}$ at $10 °\mathrm{C}$. Now, the end $P$ is maintained at $10 °\mathrm{C},$ while the end $S$ is heated and maintained at $400 °\mathrm{C}.$ The system is thermally insulated from its surroundings. If the thermal conductivity of wire $PQ$ is twice that of the wire $RS$ and the coefficient of linear thermal expansion of $PQ$ is $1.2\times {10}^{-5} {\mathrm{K}}^{-1}$, the change in length of the wire $PQ$ is