Series and Parallel Connection of Rods

An amount $n$ (in moles) of a monatomic gas at an initial temperature $T_0$ is enclosed in a cylindrical vessel fitted with a light piston. The surrounding air has a temperature $T_s(>T_0)$ and the atmospheric pressure is $p_a$. Heat may be conducted between the surrounding and the gas through the bottom of the cylinder. The bottom has a surface area $A$, thickness $x$ and thermal conductivity $K$. Assuming all changes to be slow, find the distance moved by the piston in time $t$.

Two bodies of masses $m_1$ and $m_2$ and specific heat capacities $s_1$ and $s_2$ are connected by a rod of length $L$, cross-sectional area $A$, thermal conductivity $K$ and negligible heat capacity. The whole system is thermally insulated. At time $t=0$, the temperature of the first body is $T_1$ and the temperature of the second body is ${\mathrm{T}}_2\left({\mathrm{T}}_2>{\mathrm{T}}_1\right)$. Find the temperature difference between the two bodies at time $t$.

A hollow metallic sphere of radius $20 \mathrm{cm}$ surrounds a concentric metallic sphere of radius $5 \mathrm{cm}$. The space between the two-sphere is filled with nonmetallic material. The inner and outer spheres are maintained at $50 °\mathrm{C}$ and $10 °\mathrm{C}$, respectively, and it is found that $100 \mathrm{J}$ of heat passes from the inner sphere to the outer sphere per second. Find the thermal conductivity of the material between the spheres.

A rod of negligible heat capacity has length $20 \mathrm{cm}$, area of cross section $1.0 {\mathrm{cm}}^2$ and thermal conductivity $200 {\mathrm{Wm}}^{-1 }°{\mathrm{C}}^{-1}$. The temperature of one end is maintained at zero degree Celsius and that of other end is linearly varied from $0 °\mathrm{C}$ to $60 °\mathrm{C}$ in $10$ $\mathrm{minutes}$. Assuming no loss of heat through the sides, find the total heat transmitted through the rod in these $10$ $\mathrm{minutes}$.

Seven rods $A, B, C, D, E, F \mathrm{and} G$ are joined as shown in figure. All the rods have an equal cross-sectional area $A$ and length $l$. The thermal conductivities of the rods are $K_A=K_C=K_0, K_B=K_D=2K_0, K_E=3K_0, K_F=4K_0$ and $K_G=5K_0$. The rod $E$ is kept at a constant temperature $T_1$ and the rod $G$ is kept at a constant temperature $T_2\left(T_2>T_1\right)$. (a) Show that the rod $F$ has a uniform temperature $T=\left(T_1+2T_2\right)/3$. (b) Find the rate of heat flowing from the source which maintains the temperature $T_2$.

Four identical rods $AB, CD ,CF \mathrm{and} DE$ are joined as shown in the figure. The length, cross-sectional area and thermal conductivity of each rod are $l$, $A \mathrm{and} K$, respectively. The ends $A, E \mathrm{and} F$ are maintained at temperatures $T_1, T_2 \mathrm{and} T_3$, respectively. Assuming no loss of heat to the atmosphere, find the temperature at $B$.

The three rods shown in the figure have identical geometrical dimensions. Heat flows from the hot end at a rate of $40 \mathrm{W}$ in the arrangement (a). Find the rates of heat flow when the rods are joined as in arrangement (b) and in (c). Thermal conductivities of aluminium and copper are $200 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$ and $400 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$, respectively.

The two rods shown in the figure have identical geometrical dimensions. They are in contact with two heat baths at temperatures $100 °\mathrm{C}$ and $0 °\mathrm{C}$. The temperature of the junction is $70 °\mathrm{C}$. Find the temperature of the junction if the rods are interchanged.

A room has a window fitted with single $1.0 \mathrm{m}\times 2.0 \mathrm{m}$ glass of thickness $2 \mathrm{mm}$. (a) Calculate the rate of heat flow through the closed window when the temperature inside the room is $32 °\mathrm{C}$ and that outside is $40 °\mathrm{C}$. (b) The glass is now replaced by two glass panels, each having a thickness of $1 \mathrm{mm}$ and separated by a distance of $1 \mathrm{mm}$. Calculate the rate of heat flow under the same conditions of temperature. Thermal conductivity of window glass$=1\cdot 0 \mathrm{J} {\mathrm{s}}^{-1} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$ and that of air$=0\cdot 025 \mathrm{J} {\mathrm{s}}^{-1 }{\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$.

Suppose the bent part of the frame of previous problem has a thermal conductivity of $780 \mathrm{J} {\mathrm{s}}^{-1} \mathrm{m} °{\mathrm{C}}^{-1}$, whereas, it is $390 \mathrm{J} {\mathrm{s}}^{-1} \mathrm{m} °{\mathrm{C}}^{-1}$ for the straight part. Calculate the ratio of the rate of heat flow through the bent part to the rate of heat flow through the straight part.

Consider the situation shown in figure. The frame is made of the same material and has a uniform cross-sectional area everywhere. Calculate the amount of heat flowing per second through a cross-section of the bent part if the total heat taken out per second from the end at $100 °\mathrm{C}$ is $130 \mathrm{J}$.

Figure  shows an aluminum rod joined to a copper rod. Each of the rods has a length of $20 \mathrm{cm}$ and area of cross-section $0·20 {\mathrm{cm}}^2$. The junction is maintained at a constant temperature $40 °\mathrm{C}$ and the two ends are maintained at $80 °\mathrm{C}$. Calculate the amount of heat taken out from the cold junction in one minute after the steady-state is reached. The conductivities are $K_{\mathrm{At}}=200 \mathrm{W} {\mathrm{m}}^{-1 }°{\mathrm{C}}^{-1}$ and $K_{\mathrm{Cu}}=400 \mathrm{W} {\mathrm{m}}^{-1 }°{\mathrm{C}}^{-1}$.

 An aluminium rod and a copper rod of equal length $1.0 \mathrm{m}$ and cross-sectional area $1 {\mathrm{cm}}^2$ are welded together as shown in the figure. One end is kept at a temperature of $20 °\mathrm{C}$ and the other at $60 °\mathrm{C}$. Calculate the amount of heat taken out per second from the hot end. Thermal conductivity of aluminium$=200 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$ and of copper$=390 \mathrm{W} {\mathrm{m}}^{-1} °{\mathrm{C}}^{-1}$.

The figure shows a copper rod joined to a steel rod. The rods have equal length and equal cross-sectional area. The free end of the copper rod is kept at $0 °\mathrm{C}$ and that of the steel rod is kept at  $100°\mathrm{C}$. Find the temperature at the junction of the rods. The conductivity of copper$=390 {\mathrm{Wm}}^{-1}°{\mathrm{C}}^{-1}$ and that of steel$=46 {\mathrm{Wm}}^{-1 }°{\mathrm{C}}^{-1}$.

A composite slab is prepared by pasting two plates of thickness $L_1 \mathrm{and} L_2$ and thermal conductivities $K_1 \mathrm{and} K_2$. The slabs have equal cross-sectional area. Find the equivalent conductivity of the composite slab.