Concepts of Physics 2 for IIT JEE

H C Verma Harish Chandra Verma

Chapter 28

Heat Transfer - all with Video Answers

Educators

Chapter Questions

Problem 1

A uniform slab of dimension $10 \mathrm{~cm} \times 10 \mathrm{~cm} \times 1 \mathrm{~cm}$ is kept between two heat reservoirs at temperatures $10^{\circ} \mathrm{C}$ and $90^{\circ} \mathrm{C}$. The larger surface areas touch the reservoirs. The thermal conductivity of the material is $0-80 \mathrm{~W} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}$. Find the amount of heat flowing through the slab per minute.

Surendra Kumar

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Problem 2

A liquid-nitrogen container is made of a $1-\mathrm{cm}$ thick styrofoam $\quad$ sheet $\quad$ having $\quad$ thermal conductivity $0025 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$. Liquid nitrogen at $80 \mathrm{~K}$ is kept in
it. A total area of $0.80 \mathrm{~m}^{2}$ is in contact with the liquid nitrogen. The atmospheric temperature is $300 \mathrm{~K}$. Calculate the rate of heat flow from the atmosphere to the liquid nitrogen.

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Problem 3

The normal body-temperature of a person is $97^{\circ} \mathrm{F}$. Calculate the rate at which heat is flowing out of his body through the clothes assuming the following values. Room temperature $=47^{\circ} \mathrm{F}$, surface of the body under clothes $=1 \cdot 6 \mathrm{~m}^{2}, \quad$ conductivity of the cloth $=0 \cdot 04 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-10} \mathrm{C}^{-1}$, thickness of the cloth $=0.5 \mathrm{~cm}$

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Problem 4

Water is boiled in a container having a bottom of surface area $25 \mathrm{~cm}^{2}$, thickness $1 \cdot 0 \mathrm{~mm}$ and thermal conductivity $50 \mathrm{~W} \mathrm{~m}^{-10} \mathrm{C}^{-1}, 100 \mathrm{~g}$ of water is converted into steam per minute in the steady state after the boiling starts. Assuming that no heat is lost to the atmosphere, calculate the temperature of the lower surface of the bottom. Latent heat of vaporization of water $=2 \cdot 26 \times 10^{6} \mathrm{~J} \mathrm{~kg}^{-1}$

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Problem 5

One end of a steel rod $\left(K=46 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}\right)$ of length $1 \cdot 0 \mathrm{~m}$ is kept in ice at $0^{\circ} \mathrm{C}$ and the other end is kept in boiling water at $100^{\circ} \mathrm{C}$. The area of cross section of the rod is $0-04 \mathrm{~cm}^{2}$. Assuming no heat loss to the atmosphere, find the mass of the ice melting per second. Latent heat of fusion of ice $=3.36 \times 10^{\mathrm{s}} \mathrm{J} \mathrm{kg}^{-1}$.

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Problem 6

An icebox almost completely filled with ice at $0^{\circ} \mathrm{C}$ is dipped into a large volume of water at $20^{\circ} \mathrm{C}$. The box has walls of surface area $2400 \mathrm{~cm}^{2}$, thickness $2 \cdot 0 \mathrm{~mm}$ and thermal conductivity $0.06 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$. Calculate the rate at which the ice melts in the box. Latent heat of fusion of ice $=3 \cdot 4 \times 10^{-1} \mathrm{~J} \mathrm{~kg}^{-1}$.

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Problem 7

A pitcher with $1-\mathrm{mm}$ thick porous walls contains $10 \mathrm{~kg}$ of water. Water comes to its outer surface and evaporates at the rate of $0 \cdot 1 \mathrm{~g} \mathrm{~s}^{-1} .$ The surface area of the pitcher (one side) $=200 \mathrm{~cm}^{2}$. The room temperature $=42^{\circ} \mathrm{C}$, latent heat of vaporization $=2-27 \times 10^{6} \mathrm{~J} \mathrm{~kg}^{-1}$, and the thermal conductivity of the porous walls $=0.80$ $\mathrm{J} \mathrm{s}^{-1} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1} .$ Calculate the temperature of water in the pitcher when it attains a constant value.

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Problem 8

A steel frame $\left(K=45 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}\right)$ of total length $60 \mathrm{~cm}$ and cross sectional area $0-20 \mathrm{~cm}^{2}$, forms three sides of a square. The free ends are maintained at $20^{\circ} \mathrm{C}$ and $40^{\circ} \mathrm{C}$. Find the rate of heat flow through a cross section of the frame.

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Problem 9

Water at $50^{\circ} \mathrm{C}$ is filled in a closed cylindrical vessel of height $10 \mathrm{~cm}$ and cross sectional area $10 \mathrm{~cm}^{2}$. The walls of the vessel are adiabatic but the flat parts are made of $1-\mathrm{mm} \quad$ thick $\quad$ aluminium $\quad\left(K=200 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}\right)$. Assume that the outside temperature is $20^{\circ} \mathrm{C}$. The density of water is $1000 \mathrm{~kg} \mathrm{~m}^{-5}$, and the specific heat capacity of water $=4200 \mathrm{~J} \mathrm{k} \stackrel{-\mathrm{l}}{\mathrm{g}} \mathrm{C}^{-1}$. Estimate the time taken for the temperature to fall by $1 \cdot 0^{\circ} \mathrm{C}$. Make any simplifying assumptions you need but specify them.

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Problem 10

The left end of a copper rod (length $=20 \mathrm{~cm}$, area of cross section $\left.=0-20 \mathrm{~cm}^{2}\right)$ is maintained at $20^{\circ} \mathrm{C}$ and the right end is maintained at $80^{\circ} \mathrm{C}$. Neglecting any loss of heat through radiation, find (a) the temperature at a point $11 \mathrm{~cm}$ from the left end and (b) the heat current through the rod. Thermal conductivity of copper $=385 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$.

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Problem 11

The ends of a metre stick are maintained at $100^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$. One end of a rod is maintained at $25^{\circ} \mathrm{C}$. Where should its other end be touched on the metre stick so that there is no heat current in the rod in steady state?

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Problem 12

A cubical box of volume $216 \mathrm{~cm}^{3}$ is made up of $0.1 \mathrm{~cm}$ thick wood. The inside is heated electrically by a $100 \mathrm{~W}$ heater. It is found that the temperature difference between the inside and the outside surface is $5^{\circ} \mathrm{C}$ in steady state. Assuming that the entire electrical energy spent appears as heat, find the thermal conductivity of the material of the box.

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Problem 13

Figure (28-E1) shows water in a container having $2 \cdot 0 \cdot \mathrm{mm}$ thick walls made of a material of thermal conductivity $0.50 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$. The container is kept in a melting-ice bath at $0^{\circ} \mathrm{C}$. The total surface area in contact with water is $0.05 \mathrm{~m}^{2}$. A wheel is clamped inside the water and is coupled to a block of mass $M$ as shown in the figure. As the block goes down, the wheel rotates. It is found that after some time a steady state is reached in which the block goes down with a constant speed of $10 \mathrm{~cm} \mathrm{~s}^{-1}$ and the temperature of the water remains constant at $1 \cdot 0^{\circ} \mathrm{C}$. Find the mass $M$ of the block. Assume that the heat flows out of the water only through the walls in contact. Take $g=10 \mathrm{~m} \mathrm{~s}^{-2}$.

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Problem 14

On a winter day when the atmospheric temperature drops to $-10^{\circ} \mathrm{C}$, ice forms on the surface of a lake.
(a) Calculate the rate of increase of thickness of the ice when $10 \mathrm{~cm}$ of ice is already formed. (b) Calculate the total time taken in forming $10 \mathrm{~cm}$ of ice. Assume that the temperature of the entire water reaches $0^{\circ} \mathrm{C}$ before the ice starts forming. Density of water $=1000 \mathrm{~kg} \mathrm{~m}^{-3}$, latent heat of fusion of ice $=3.36 \times 10^{5} \mathrm{~J} \mathrm{~kg}^{-1}$ and thermal conductivity of ice $=1.7 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$. Neglect the expansion of water on freezing.

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Problem 15

Consider the situation of the previous problem. Assume that the temperature of the water at the bottom of the lake remains constant at $4^{\circ} \mathrm{C}$ as the ice forms on the surface (the heat required to maintain the temperature of the bottom layer may come from the bed of the lake). The depth of the lake is $1 \cdot 0 \mathrm{~m}$. Show that the thickness of the ice formed attains a steady state maximum value. Find this value. The thermal conductivity of water $=0.50 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1} .$ Take other relevant data from the previous problem.

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Problem 16

Three rods of lengths $20 \mathrm{~cm}$ each and area of cross section $1 \mathrm{~cm}^{2}$ are joined to form a triangle $A B C$. The conductivities of the rods are $K_{A B}=50 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-10} \mathrm{C}^{-1}$, $K_{B C}=200 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$ and $K_{A C}=400 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$. The junctions $A, B$ and $C$ are maintained at $40^{\circ} \mathrm{C}, 80^{\circ} \mathrm{C}$ and $80^{\circ} \mathrm{C}$ respectively. Find the rate of heat flowing through the rods $A B, A C$ and $B C$

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Problem 17

A semicircular rod is joined at its end to a straight rod of the same material and the same cross-sectional area. The straight rod forms a diameter of the other rod. The junctions are maintained at different temperatures. Find the ratio of the heat transferred through a cross section of the semicircular rod to the heat transferred through a cross section of the straight rod in a given time.

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Problem 18

A metal rod of cross sectional area $1 \cdot 0 \mathrm{~cm}^{2}$ is being heated at one end. At one time, the temperature gradient is $5-0^{\circ} \mathrm{C} \mathrm{cm}^{-1}$ at cross section $A$ and is $2 \cdot 5^{\circ} \mathrm{Ccm}^{-1}$ at cross section $B$. Calculate the rate at which the temperature is increasing in the part $A B$ of the rod. The heat capacity of the part $A B=0-40 \mathrm{~J}^{\circ} \mathrm{C}^{-1}$, thermal conductivity of the material of the $\operatorname{rod}=200 \mathrm{~W} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}$. Neglect any loss of heat to the atmosphere.

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Problem 19

Steam at $120^{\circ} \mathrm{C}$ is continuously passed through a $50-\mathrm{cm}$ long rubber tube of inner and outer radii $1 \cdot 0 \mathrm{~cm}$ and $1.2 \mathrm{~cm}$. The room temperature is $30^{\circ} \mathrm{C}$. Calculate the rate of heat flow through the walls of the tube. Thermal conductivity of rubber $=0 \cdot 15 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$

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Problem 21

A hollow tube has a length $l$, inner radius $R_{1}$ and outer radius $R_{2}$. The material has a thermal conductivity $K$. Find the heat flowing through the walls of the tube if
(a) the flat ends are maintained at temperatures $T_{1}$ and $T_{2}\left(T_{2}>T_{1}\right)$ (b) the inside of the tube is maintained at temperature $T_{1}$ and the outside is maintained at $T_{2}$

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Problem 22

A composite slab is prepared by pasting two plates of thicknesses $L_{1}$ and $L_{2}$ and thermal conductivities $K_{1}$ and $K_{2} .$ The slabs have equal cross-sectional area. Find the equivalent conductivity of the composite slab.

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Problem 23

Figure (28-E2) 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 $0^{\circ} \mathrm{C}$ and that of the steel rod is kept at $100^{\circ} \mathrm{C}$. Find the temperature at the junction of the rods. Conductivity of copper $=390 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$ and that of steel $=46 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$.

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Problem 24

An aluminium rod and a copper rod of equal length $1 \cdot 0 \mathrm{~m}$ and cross-sectional area $1 \mathrm{~cm}^{2}$ are welded together as shown in figure (28-E3). One end is kept at a temperature of $20^{\circ} \mathrm{C}$ and the other at $60^{\circ} \mathrm{C} .$ Calculate the amount of heat taken out per second from the hot end. Thermal conductivity of aluminium $=200 \mathrm{Wm}^{-1}{ }^{\circ} \mathrm{C}^{-1}$ and of copper $=390 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$

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Problem 25

Figure (28-E4) shows an aluminium rod joined to a copper rod. Each of the rods has a length of $20 \mathrm{~cm}$ and area of cross section $020 \mathrm{~cm}^{2}$. The junction is maintained at a constant temperature $40^{\circ} \mathrm{C}$ and the two ends are maintained at $80^{\circ} \mathrm{C}$. Calculate the amount of heat taken out from the cold junction in one minute after the steady state is reached. The conductivities are $K_{\mu}=200 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$ and $K_{c u}=400 \mathrm{~W} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$

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Problem 26

Consider the situation shown in figure (28-E5). 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 $100^{\circ} \mathrm{C}$ is $130 \mathrm{~J}$.

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Problem 27

Suppose the bent part of the frame of the previous $\begin{array}{llll}\text { problem } & \text { has } & \text { a } & \text { thermal } & \text { conductivity of }\end{array}$ $780 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$ whereas it is $390 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-1_{0}} \mathrm{C}^{-1}$ 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.

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Problem 28

A room has a window fitted with a single $1-0 \mathrm{~m} \times 2 \cdot 0 \mathrm{~m}$ glass of thickness $2 \mathrm{~mm}$. (a) Calculate the rate of heat flow through the closed window when the temperature inside the room is $32^{\circ} \mathrm{C}$ and that outside is $40^{\circ} \mathrm{C} .$ (b) The glass is now replaced by two glasspanes, 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 $=1 \cdot 0 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-10} \mathrm{C}^{-1}$ and that of air
$=0 \cdot 025 \mathrm{~J} \mathrm{~s}^{-1} \mathrm{~m}^{-10} \mathrm{C}^{-1}$.

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Problem 29

The two rods shown in figure $(28-\mathrm{E} 6)$ have identical geometrical dimensions. They are in contact with two heat baths at temperatures $100^{\circ} \mathrm{C}$ and $0^{\circ} \mathrm{C}$. The temperature of the junction is $70^{\circ} \mathrm{C}$. Find the temperature of the junction if the rods are interchanged.

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Problem 30

The three rods shown in figure $(28-\mathrm{E} 7)$ 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 $200 \mathrm{~W} \mathrm{~m}^{-1}{ }^{\circ} \mathrm{C}^{-1}$ and $400 \mathrm{~W} \mathrm{~m}^{-1}{ }^{-1} \mathrm{C}^{-1}$
respectively.

Khoobchandra Agrawal

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Problem 31

Four identical rods $A B, C D, C F$ and $D E$ are joined as shown in figure $(28-\mathrm{E} 8)$. The length, cross-sectional area and thermal conductivity of each rod are $l, A$ and $K$ respectively. The ends $A, E$ and $F$ are maintained at temperatures $T_{1}, T_{2}$ and $T_{3}$ respectively. Assuming no loss of heat to the atmosphere, find the temperature at $B$.

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Problem 32

Seven rods $A, B, C, D, E, F$ and $G$ are joined as shown in figure (28-E9). All the rods have equal cross-sectional area $A$ and length $l$. The thermal conductivities of the rods are $K_{A}=K_{C}=K_{0}, K_{B}=K_{D}=2 K_{0}, K_{E}=3 K_{0}, K_{F}=4 K_{0}$
and $K_{G}=5 K_{0-}$ 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 $T=\left(T_{1}+2 T_{2}\right) / 3$
(b) Find the rate of heat flowing from the source which maintains the temperature $T_{2}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

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Problem 33

Find the rate of heat flow through a cross section of the rod shown in figure $(28-\mathrm{E} 10) \quad\left(\theta_{2}>\theta_{1}\right) . \quad$ Thermal conductivity of the material of the rod is $K$.

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Problem 34

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

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Problem 35

A hollow metallic sphere of radius $20 \mathrm{~cm}$ surrounds a concentric metallic sphere of radius $5 \mathrm{~cm}$. The space between the two spheres is filled with a nonmetallic material. The inner and outer spheres are maintained at $50^{\circ} \mathrm{C}$ and $10^{\circ} \mathrm{C}$ 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.

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Problem 36

Figure (28-E11) shows two adiabatic vessels, each containing a mass $m$ of water at different temperatures. The ends of a metal rod of length $L$, area of cross section $A$ and thermal conductivity $K$, are inserted in the water as shown in the figure. Find the time taken for the difference between the temperatures in the vessels to become half of the original value. The specific heat capacity of water is $s$. Neglect the heat capacity of the rod and the container and any loss of heat to the atmosphere.

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Problem 37

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 $T_{2}\left(T_{2}>T_{1}\right)$. Find the temperature difference between the two bodies at time $t$.

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Problem 38

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 $T_{s}\left(>T_{0}\right)$ 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 $t$.

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Problem 39

Assume that the total surface area of a human body is $1-6 \mathrm{~m}^{2}$ and that it radiates like an ideal radiator. Calculate the amount of energy radiated per second by the body if the body temperature is $37^{\circ} \mathrm{C}$. Stefan constant $\sigma$ is $6 \cdot 0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$.

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Problem 40

Calculate the amount of heat radiated per second by a body of surface area $12 \mathrm{~cm}^{2}$ kept in thermal equilibrium in a room at temperature $20^{\circ} \mathrm{C}$. The emissivity of the surface $=0 \cdot 80$ and $\sigma=6 \cdot 0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$

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Problem 41

A solid aluminium sphere and a solid copper sphere of twice the radius are heated to the same temperature and are allowed to cool under identical surrounding temperatures. Assume that the emissivity of both the spheres is the same. Find the ratio of (a) the rate of heat loss from the aluminium sphere to the rate of heat loss from the copper sphere and $(\mathrm{b})$ the rate of fall of temperature of the aluminium sphere to the rate of fall of temperature of the copper sphere. The specific heat capacity of aluminium $=900 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{\circ} \mathrm{C}^{-1}$ and that of copper $=390 \mathrm{~J} \mathrm{~kg}^{-1}{ }^{-1} \mathrm{C}^{-1}$. The density of copper $=3-4$ times the density of aluminium.

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Problem 42

A $100 \mathrm{~W}$ bulb has tungsten filament of total length $1.0 \mathrm{~m}$ and radius $4 \times 10^{-1} \mathrm{~m}$. The emissivity of the filament is $0-8$ and $\sigma=6-0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}$. Calculate the
temperature of the filament when the bulb is operating at correct wattage.

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Problem 43

A spherical ball of surface area $20 \mathrm{~cm}^{2}$ absorbs any radiation that falls on it. It is suspended in a closed box maintained at $57^{\circ} \mathrm{C}$. (a) Find the amount of radiation falling on the ball per second. (b) Find the net rate of heat flow to or from the ball at an instant when its temperature is $200^{\circ} \mathrm{C}$. Stefan constant $=6.0 \times 10^{-8}$ $\mathrm{W} \mathrm{m}^{-2} \mathrm{~K}^{-4}$

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Problem 44

A spherical tungsten piece of radius $1 \cdot 0 \mathrm{~cm}$ is suspended in an evacuated chamber maintained at $300 \mathrm{~K}$. The piece is maintained at $1000 \mathrm{~K}$ by heating it electrically. Find the rate at which the electrical energy must be supplied. The emissivity of tungsten is $0: 30$ and the Stefan constant $\sigma$ is $6.0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$

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Problem 45

A cubical block of mass $1 \cdot 0 \mathrm{~kg}$ and edge $5 \cdot 0 \mathrm{~cm}$ is heated to $227^{\circ} \mathrm{C}$. It is kept in an evacuated chamber maintained at $27^{\circ} \mathrm{C}$. Assuming that the block emits radiation like a blackbody, find the rate at which the temperature of the block will decrease. Specific heat capacity of the material of the block is $400 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$.

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Problem 46

A copper sphere is suspended in an evacuated chamber maintained at $300 \mathrm{~K}$. The sphere is maintained at a constant temperature of $500 \mathrm{~K}$ by heating it electrically. A total of $210 \mathrm{~W}$ of electric power is needed to do it. When the surface of the copper sphere is completely blackened, $700 \mathrm{~W}$ is needed to maintain the same temperature of the sphere. Calculate the emissivity of copper.

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Problem 47

A spherical ball $A$ of surface area $20 \mathrm{~cm}^{2}$ is kept at the centre of a hollow spherical shell $B$ of area $80 \mathrm{~cm}^{2}$. The surface of $A$ and the inner surface of $B$ emit as blackbodies. Both $A$ and $B$ are at $300 \mathrm{~K}$. (a) How much is the radiation energy emitted per second by the ball $A ?$ (b) How much is the radiation energy emitted per second by the inner surface of $B$ ? (c) How much of the energy emitted by the inner surface of $B$ falls back on this surface itself ?

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Problem 48

A cylindrical rod of length $50 \mathrm{~cm}$ and cross sectional area $1 \mathrm{~cm}^{2}$ is fitted between a large ice chamber at $0^{\circ} \mathrm{C}$ and an evacuated chamber maintained at $27^{\circ} \mathrm{C}$ as shown in figure (28.E12). Only small portions of the rod are inside the chambers and the rest is thermally insulated from the surrounding. The cross section going into the evacuated chamber is blackened so that it completely absorbs any radiation falling on it. The temperature of the blackened end is $17^{\circ} \mathrm{C}$ when steady state is reached. Stefan constant $\sigma=6 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$. Find the
thermal conductivity of the material of the rod.

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Problem 49

One end of a rod of length $20 \mathrm{~cm}$ is inserted in a furnace at $800 \mathrm{~K}$. The sides of the rod are covered with an insulating material and the other end emits radiation like a blackbody. The temperature of this end is $750 \mathrm{~K}$ in the steady state. The temperature of the surrounding air is $300 \mathrm{~K}$. Assuming radiation to be the only important mode of energy transfer between the surrounding and the open end of the rod, find the thermal conduetivity of the rod. Stefan constant $\sigma=6 \cdot 0 \times 10^{-8} \mathrm{~W} \mathrm{~m}^{-2} \mathrm{~K}^{-4}$.

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Problem 50

A calorimeter of negligible heat capacity contains $100 \mathrm{cc}$ of water at $40^{\circ} \mathrm{C}$. The water cools to $35^{\circ} \mathrm{C}$ in 5 minutes. The water is now replaced by $\mathrm{K}$ -oil of equal volume at $40^{\circ} \mathrm{C}$. Find the time taken for the temperature to become $35^{\circ} \mathrm{C}$ under similar conditions. Specific heat capacities of water and $\quad$ K-oil are $4200 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}$ and $\begin{array}{llll}2100 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1} & \text { respectively. } & \text { Density of } \mathrm{K} \text { -oil }\end{array}$ $=800 \mathrm{~kg} \mathrm{~m}^{-3}$

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Problem 51

A body cools down from $50^{\circ} \mathrm{C}$ to $45^{\circ} \mathrm{C}$ in 5 minutes and to $40^{\circ} \mathrm{C}$ in another 8 minutes. Find the temperature of the surrounding.

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Problem 52

A calorimeter contains $50 \mathrm{~g}$ of water at $50^{\circ} \mathrm{C}$. The temperature falls to $45^{\circ} \mathrm{C}$ in 10 minutes. When the calorimeter contains $100 \mathrm{~g}$ of water at $50^{\circ} \mathrm{C}$, it takes 18 minutes for the temperature to become $45^{\circ} \mathrm{C}$. Find the water equivalent of the calorimeter.

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Problem 53

A metal ball of mass $1 \mathrm{~kg}$ is heated by means of a $20 \mathrm{~W}$ heater in a room at $20^{\circ} \mathrm{C}$. The temperature of the ball becomes steady at $50^{\circ} \mathrm{C}$.
(a) Find the rate of loss of heat to the surrounding when the ball is at $50^{\circ} \mathrm{C}$.
(b) Assuming Newton's law of cooling, calculate the rate of loss of heat to the surrounding when the ball is at $30^{\circ} \mathrm{C}$.
(c) Assume that the temperature of the ball rises uniformly from $20^{\circ} \mathrm{C}$ to $30^{\circ} \mathrm{C}$ in 5 minutes. Find the total loss of heat to the surrounding during this period.
(d) Calculate the specific heat capacity of the metal.

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Numerade Educator

Problem 54

A metal block of heat capacity $80 \mathrm{~J}^{\circ} \mathrm{C}^{-1}$ placed in a room at $20^{\circ} \mathrm{C}$ is heated electrically. The heater is switched off when the temperature reaches $30^{\circ} \mathrm{C}$. The temperature of the block rises at the rate of $2{ }^{\circ} \mathrm{C} \mathrm{s}^{-1}$ just after the heater is switched on and falls at the rate of $0 \cdot 2^{\circ} \mathrm{C} \mathrm{s}^{-1}$ just after the heater is switched off. Assume Newton's law of cooling to hold. (a) Find the power of the heater.
(b) Find the power radiated by the block just after the heater is switched off. (c) Find the power radiated by the block when the temperature of the block is $25^{\circ} \mathrm{C}$. (d) Assuming that the power radiated at $25^{\circ} \mathrm{C}$ represents the average value in the heating process, find the time for which the heater was kept on.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 55

A hot body placed in a surrounding of temperature $\theta_{0}$ obeys Newton's law of cooling $\frac{d \theta}{d t}=-k\left(\theta-\theta_{0}\right) .$ Its temperature at $t=0$ is $\theta_{1} .$ The specific heat capacity of the body is $s$ and its mass is $m$. Find (a) the maximum heat that the body can lose and (b) the time starting from $t=0$ in which it will lose $90 \%$ of this maximum heat.

Surendra Kumar

Numerade Educator