D. C. Pandey Solutions for Chapter: Calorimetry and Heat Transfer, Exercise 3: Exercises

A copper cube of mass $200 \mathrm{g}$ slides down a rough inclined plane of inclination $37°$ at a constant speed. Assuming that the loss in mechanical energy goes into the copper block as thermal energy, find the increase in temperature of the block as it slides down through $60 \mathrm{cm}$. Specific heat capacity of copper is equal to $420 \mathrm{J} {\mathrm{kg}}^{-1} {\mathrm{K}}^{-1}$. (Take $\mathrm{g}=10 \mathrm{m} {\mathrm{s}}^{-2}$)

An electric heater is placed inside a room of total wall area $137 {\mathrm{m}}^2$ to maintain the temperature inside at $20 °\mathrm{C}$. The outside temperature is $-10 °\mathrm{C}$. The walls are made of three composite materials. The innermost layer is made of wood of thickness $2.5 \mathrm{cm}$, the middle layer is of cement of thickness $1 \mathrm{cm}$ and the exterior layer is of brick of thickness $2.5 \mathrm{cm}$. Find the power of electric heater, assuming that there are no heat losses through the floor and the ceiling. The thermal conductivities of wood, cement and brick are $0.125 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}, 1.5 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$ and $1.0 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$, respectively.

A $2 \mathrm{m}$ long wire of resistance $4 \mathrm{\Omega }$ and diameter $0.64 \mathrm{mm}$ is coated with plastic insulation of thickness $0.06 \mathrm{mm}$. A current of $5 \mathrm{A}$ flows through the wire. Find the temperature difference across the insulation in the steady-state. Thermal conductivity of plastic is $0.16\times {10}^{-2} \mathrm{cal} {\mathrm{s}}^{-1} {\mathrm{cm}}^{-1} °{\mathrm{C}}^{-1}$.

Two chunks of metal with heat capacities $C_1$and $C_2$ are interconnected by a rod of length $l$ and cross-sectional area $A$ and fairly low conductivity $k$. The whole system is thermally insulated from the environment. At a moment $t=0$, the temperature difference between the two chunks of metal equals $(\mathrm{\Delta }T{)}_0$. Assuming the heat capacity of the rod to be negligible, find the temperature difference between the chunks as a function of time.

A rod of length $l$ with thermally insulated lateral surface, consists of a material whose heat conductivity coefficient varies with temperature as $k=\frac{a}{T}$, where $a$ is a constant. The ends of the rod are kept at temperatures $T_1$ and $T_2$. Find the function $T\left(x\right)$, where $x$ is the distance from the end whose temperature is $T_1$.

One end of a uniform brass rod $20 \mathrm{cm}$ long and $10 {\mathrm{cm}}^2$ cross-sectional area, is kept at $100 °\mathrm{C}$. The other end is in perfect thermal contact with another rod of identical cross-section and length $10 \mathrm{cm}$. The free end of this rod is kept in melting ice and when the steady-state has been reached, it is found that $360 \mathrm{g}$ of ice melts per hour. Calculate the thermal conductivity of the rod, given that the thermal conductivity of brass is $0.25 \mathrm{cal} {\mathrm{s}}^{-1} {\mathrm{cm}}^{-1} °{\mathrm{C}}^{-1}$and $L=80 \mathrm{cal} {\mathrm{g}}^{-1}$.

Heat flows radially outward through a spherical shell of outside radius $R_2$ and inner radius $R_1$ The temperature of the inner surface of the shell is ${\mathrm{\theta }}_1$ and that of outer is ${\mathrm{\theta }}_2$. At what radial distance from the centre of the shell, the temperature is just half-way between ${\mathrm{\theta }}_1$ and ${\mathrm{\theta }}_2$?

A layer of ice of thickness $y$ is on the surface of a lake. The air is at a constant temperature $-\mathrm{\theta } °\mathrm{C}$ and the ice water interface is at $0 °\mathrm{C}$. Show that the rate at which the thickness increases is given by

$\frac{\mathrm{d}y}{\mathrm{d}t}=\frac{K\mathrm{\theta }}{L\rho y}$

where, $K$ is the thermal conductivity of the ice, $L$ is the latent heat of fusion and $\rho$ is the density of ice.