A Complete Resource Book in Physics for JEE Main

Sanjeev Kumar

Chapter 11

Heat Transfer - all with Video Answers

Educators

RZ

Chapter Questions

Problem 1

The ends of a uniform metre stick of iron are maintained at $80^{\circ} \mathrm{C}$ and $30^{\circ} \mathrm{C}$. One end of another rod is maintained at $50^{\circ} \mathrm{C}$, where its other end should be touched on the metre stick so that there is no heat current in the rod in steady state?
(A) $40 \mathrm{~cm}$ from hot end
(B) $40 \mathrm{~cm}$ from cold end
(C) $50 \mathrm{~cm}$ from cold end
(D) $70 \mathrm{~cm}$ from cold end

Sunita Kumari

Sunita Kumari

Numerade Educator

Problem 2

A body cools from $60^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$ in $10 \mathrm{~min}$. If the room temperature is $25^{\circ} \mathrm{C}$ and assuming Newton's law of cooling to hold good, the temperature of the body at the end of the next 10 min will be
(A) $38.5^{\circ} \mathrm{C}$
(B) $40^{\circ} \mathrm{C}$
(C) $42.85^{\circ} \mathrm{C}$
(D) $45^{\circ} \mathrm{C}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 3

Two rods of equal length and area of cross-section are kept parallel and lagged between temperature $20^{\circ} \mathrm{C}$ and $80^{\circ} \mathrm{C}$. The ratio of the effective thermal conductivity to that of the first rod is $\left(K_{1} / K_{2}\right)=3: 4$
(A) $7: 4$
(B) $7: 6$
(C) $4: 7$
(D) $7: 8$

Narayan Hari

Numerade Educator

Problem 4

If the temperature of the sun is increased from $T$ to $2 T$ and its radius from $R$ to $2 R$, then the ratio of the radiant energy received on earth to what it was previously will be
(A) 4
(B) 16
(C) 32
(D) 64

Narayan Hari

Numerade Educator

Problem 5

A hot body is being cooled in air according to Newton's law of cooling, the rate of fall of temperature being $k$ times the difference of its temperature with respect to that of surroundings. The time, after which the body will lose half the maximum heat it can lose, is
(A) $\frac{1}{k}$
(B) $\frac{\ln 2}{k}$
(C) $\frac{\ln 3}{k}$
(D) $\frac{2}{k}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 6

A block body is at a temperature $2880 \mathrm{~K}$. The energy radiation emitted by this object with wavelength between $499 \mathrm{~nm}$ and $500 \mathrm{~nm}$ is $U_{1}$, between $999 \mathrm{~mm}$ and $1000 \mathrm{~nm}$ is $U_{2}$, and between $1499 \mathrm{~nm}$ and $1500 \mathrm{~nm}$ is $U_{3}$, then (Wien's constant $b=2.88 \times 10^{6} \mathrm{~nm}-\mathrm{K}$ )
(A) $U_{1}>U_{2}$
(B) $U_{2}>U_{1}$
(C) $U_{1}=0$
(D) $U_{3}=0$

Narayan Hari

Numerade Educator

Problem 7

On the surface of lake when the atmospheric temperature is $-15^{\circ} \mathrm{C}, 1.5 \mathrm{~cm}$ thick layer of ice is formed in $20 \mathrm{~min}$, time taken to change its thickness from $1.5 \mathrm{~cm}$ to $3 \mathrm{~cm}$ will be
(A) $20 \mathrm{~min}$
(B) less than $20 \mathrm{~min}$
(C) greater than $40 \mathrm{~min}$
(D) less than $40 \mathrm{~min}$ and greater than $20 \mathrm{~min}$

Narayan Hari

Numerade Educator

Problem 8

A diatomic molecule having atoms of masses $m_{1}$ and $m_{2}$ has its potential energy function about the equilibrium position $r_{0}$ as given by $U(r)=-A+B\left(r-r_{0}\right)^{2}$, where $A$ and $B$ are constants. When the atom vibrate at high temperature condition, the square of angular frequency of vibration will be
(A) $\frac{2 B}{m_{1}}$
(B) $\frac{2 B}{m_{2}}$
(C) $\frac{2 B\left(m_{1}+m_{2}\right)}{m_{1} m_{2}}$
(D) $\frac{B\left(m_{1}+m_{2}\right)}{2 m_{1} m_{2}}$

Narayan Hari

Numerade Educator

Problem 9

It is known that the temperature in the room is $+20^{\circ} \mathrm{C}$ when the outdoor temperature is $-20^{\circ} \mathrm{C}$ and $+10^{\circ} \mathrm{C}$ when the outdoor temperature is $-40^{\circ} \mathrm{C}$. Then what is the temperature $T$ of the radiator heating the room? (Assuming that radiated by the heater is proportional to the temperature difference with the room.)
(A) $40^{\circ} \mathrm{C}$
(B) $60^{\circ} \mathrm{C}$
(C) $30^{\circ} \mathrm{C}$
(D) $20^{\circ} \mathrm{C}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 10

One mole of an ideal gas with heat capacity at constant pressure $C_{P}$ undergoes the process $T=T_{0}+\alpha V$, where $T_{0}$ and $\alpha$ are constants. If its volume increases from $V_{1}$ to $V_{2}$, the amount of heat transferred to the gas is
(A) $C_{P} R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(B) $\alpha C_{P} \frac{\left(V_{2}-V_{1}\right)}{R T_{0}} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(C) $\alpha C_{P}\left(V_{2}-V_{1}\right)+R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)$
(D) $R T_{0} \ln \left(\frac{V_{2}}{V_{1}}\right)+\alpha C_{P}\left(V_{1}-V_{2}\right)$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 11

Two bodies each having a heat capacity of $C=500 \mathrm{~J} / \mathrm{K}$ are joined together by a rod of length $L=40.0 \mathrm{~cm}$, thermal conductivity $20 \mathrm{~W} / \mathrm{mK}$, and cross-sectional area of $S=3.00 \mathrm{~cm}^{2}$. The bodies are joined with the help of a thermally insulated rod. The time after which temperature difference diminishes $\eta=2$ times is (Disregard the heat capacity of the rod.)
(A) $193 \mathrm{~min}$
(B) $240 \mathrm{~min}$
(C) $77 \mathrm{~min}$
(D) $144 \mathrm{~min}$

Hunza Gilgit

Numerade Educator

Problem 12

A rod of length $l$ (laterally thermally insulated) of uniform cross-sectional area $A$ consists of a material whose thermal conductivity varies with temperature as $K=\frac{K_{o}}{a+b T}$, where $K_{0}, a$ and $b$ are constants. $T_{1}$ and $T_{2}\left(<T_{1}\right)$ are the temperature of two ends of rod. Then rate of flow of heat across the rod is
(A) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{1}}{a+b T_{2}}\right)$
(B) $\frac{A K_{0}}{b l}\left(\frac{a+b T_{2}}{a+b T_{1}}\right)$
(C) $\frac{A K_{0}}{b l} \ln \left[\frac{a+b T_{1}}{a+b T_{2}}\right]$
(D) $\frac{A K_{0}}{a l} \ln \left[\frac{a+b T_{2}}{a+b T_{1}}\right]$

Narayan Hari

Numerade Educator

Problem 13

A copper sphere is suspended in an evacuated chamber maintained at $300 \mathrm{~K}$. The sphere is maintained at constant temperature of $900 \mathrm{~K}$ by heating electrically. A total of $300 \mathrm{~W}$ electric power is needed to do this. When half of the surface of the copper sphere is completely blackened, $600 \mathrm{~W}$ is needed to maintain the same temperature of sphere. The emissivity of copper is
(A) $1 / 4$
(B) $1 / 3$
(C) $1 / 2$
(D) 1

Sunita Kumari

Numerade Educator

Problem 14

A coil of resistance $R$ connected to an external battery is placed inside an adiabatic cylinder fitted with a frictionless piston and containing an ideal gas. A current $I=a_{0} t$ flows through the coil $\left(a_{0}\right.$ is a teve constant). For time interval $t=0$ to $t=t_{0}$, the piston goes up to a height of $($ Assume $\Delta \mathrm{U}=0)$
(A) $\frac{a_{0}^{2} R^{2} t_{0}^{2}}{2 m g}$
(B) $\frac{a_{0}^{2} R t_{0}^{3}}{2 m g}$
(C) $\frac{a_{0}^{2} R t_{0}^{3}}{3 m g}$
(D) $\frac{a_{0}^{2} R t_{0}^{2}}{3 m g}$

Hunza Gilgit

Numerade Educator

Problem 15

In a composite rod, when two rods of different lengths and of the same area of cross-section are joined end to end, then if $K$ is the effective coefficient of thermal conductivity, $\frac{l_{1}+l_{2}}{K}$ is equal to
(A) $\frac{l_{1}}{K_{1}}-\frac{l_{2}}{K_{2}}$
(B) $\frac{l_{1}}{K_{2}}-\frac{l_{2}}{K_{1}}$
(C) $\frac{l_{1}}{K_{1}}+\frac{l_{2}}{K_{2}}$
(D) $\frac{l_{1}}{K_{2}}+\frac{l_{2}}{K_{1}}$

Mahendra K

Numerade Educator

Problem 16

The SI unit of thermal conductivity is
(A) $\mathrm{Js}^{-1} \mathrm{~m} \mathrm{~K}^{-1}$
(B) $\mathrm{J} \mathrm{sm}^{-1} \mathrm{~K}^{-1}$
(C) $\mathrm{Jsm}^{-1} \mathrm{~K}$
(D) $\mathrm{Js}^{-1} \mathrm{~m}^{-1} \mathrm{~K}^{-1}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 17

According to Wien's displacement law,
(A) $\lambda_{m}=$ constant
(B) $\lambda_{m} T=$ constant
(C) $\lambda_{m} T^{2}=$ constant
(D) $\lambda_{m}^{2} T=$ constant

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 18

A bucket full of hot water is kept in a room and it cool from $75^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ in $T_{1}$ min, from $70^{\circ} \mathrm{C}$ to $65^{\circ} \mathrm{C}$ in $T_{2}$ $\min$ and from $65^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in $T_{3} \mathrm{~min}$. Then
(A) $T_{1}=T_{2}=T_{3}$
(B) $T_{1}<T_{2}<T_{3}$
(C) $T_{1}>T_{2}>T_{3}$
(D) $T_{1}<T_{2}>T_{3}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 19

Three rods made of the same material and having the same cross-section have been joined as shown in Fig. $11.2$. Each rod is of the same length. The left and right ends are kept at $0^{\circ} \mathrm{C}$ and $90^{\circ} \mathrm{C}$, respectively. The temperature of the junction of the three rods will be
(A) $45^{\circ} \mathrm{C}$
(B) $60^{\circ} \mathrm{C}$
(C) $30^{\circ} \mathrm{C}$
(D) $20^{\circ} \mathrm{C}$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 20

When a hollow and a solid sphere of same material with same outer radius and identical surface finish are heated to the same temperature
(A) In the beginning, both will emit equal amount of radiation per unit time
(B) In the beginning, both will absorb equal amount of radiation per unit time
(C) Both spheres will have same rate of fall of temperature $\left(\frac{d T}{d t}\right)$
(D) Both spheres will have equal temperatures at any moment

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 21

A $10 \mathrm{~g}$ body is kept in an enclosure of $27^{\circ} \mathrm{C}$. For body's temperature $127^{\circ} \mathrm{C}$, the specific heat $0.1 \mathrm{~K} \mathrm{cal} / \mathrm{kg}^{\circ} \mathrm{C}$
and surface area $10^{-3} \mathrm{~m}^{2}$. The $\left(\sigma=5.67 \times 10^{-8} \mathrm{~W} / \mathrm{m}^{2} \mathrm{k}^{4}\right]$
(A) Rate of cooling is $0.227 \mathrm{ks}^{-1}$
(B) Rate of cooling will be zero at $400 \mathrm{~K}$ enclosure
(C) Cooling does not take place
(D) Cooling will be faster at $127^{\circ} \mathrm{C}$ enclosure

Hunza Gilgit

Numerade Educator

Problem 22

The temperature drop through a two-layered furnace wall is $900^{\circ} \mathrm{C}$. Each layer is of equal area of cross-section. Which of the following actions will result in lowering the temperature $\theta$ of the interface?
(A) By increasing the thermal conductivity of outer layer
(B) By increasing the thermal conductivity of inner layer
(C) By increasing thickness of outer layer
(D) By increasing thickness of inner layer

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 23

If the temperature of the sun is increased from $T$ to $2 T$ and its radius from $R$ to $2 R$, then the ratio of the radiant energy received on earth to what it was previously will be
(A) 4
(B) 16
(C) 32
(D) 64

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 24

The curved surface of uniform rod is thermally isolated from surrounding. Its ends are maintained at temperature $T_{1}$ and $T_{2}\left(T_{1}>T_{2}\right) .$ If in steady state, temperature gradient at a distance $x$ from hot end is equal to $\frac{d T}{d x}$, then which one of the following graphs is correct?
(A)
(B)
(C)
(D)

Dheeraj Sharma

Numerade Educator

Problem 25

The rate of flow of thermal current depends on the nature of material (thermal conductivity), cross-sectional area $A$, and temperature gradient. More the temperature difference, higher the thermal current flow. This fact identifies the thermal resistance offered by the material while conducting heat. One can find by equivalent resistance in heat flow using the same principles as for current. A body may transfer energy better by radiation. The nature of radiating surfaces play a role in the power radiated from it. On covering a surface by non-conducting/radiating media, the loss of the heat energy can be controlled.
A rod of length $l$ and conductivity $k$ is placed between two reservoirs maintained at $0^{\circ} \mathrm{C}$ and $100^{\circ} \mathrm{C}$. At a distance $\frac{l}{3}$ and $\frac{l}{5}$ from $100^{\circ} \mathrm{C}$ reservoir the rate of flow of thermal current are $I_{1}$ and $I_{2} .$ The temperatures at $\frac{l}{5}$ and $\frac{l}{3}$ are $\theta_{1}$ and $\theta_{2}$ then
(A) $I_{1}=I_{2}, 2 \theta_{1}=3 \theta_{2}$
(B) $5 \theta_{1}-3 \theta_{2}=200$
(C) $I_{1}=I_{2}, 5 \theta_{1}-3 \theta_{2}=200$
(D) $I_{1} \neq I_{2}, \quad 5 \theta_{2}=3 \theta_{1}$

Sunita Kumari

Numerade Educator

Problem 26

The rate of flow of thermal current depends on the nature of material (thermal conductivity), cross-sectional area $A$, and temperature gradient. More the temperature difference, higher the thermal current flow. This fact identifies the thermal resistance offered by the material while conducting heat. One can find by equivalent resistance in heat flow using the same principles as for current. A body may transfer energy better by radiation. The nature of radiating surfaces play a role in the power radiated from it. On covering a surface by non-conducting/radiating media, the loss of the heat energy can be controlled.
Two spheres with emissive powers $0.6$ and $0.8$ of radii $2 \mathrm{~cm}$ and $4 \mathrm{~cm}$ are heated to temperatures of $27^{\circ} \mathrm{C}$ and $157^{\circ} \mathrm{C}$ and placed in a room at absolute temperature $0 \mathrm{~K}$. The ratio of heat radiated per second is
(A) $0.187$
(B) $1.6 \times 10^{-4}$
(C) $0.079$
(D) $0.831$

Sunita Kumari

Numerade Educator

Problem 27

Two rods of material $X$ and two rods of material $Y$ are placed with a rod of material $Z$ as shown with the junctions $A$ and $D$ at $100^{\circ} \mathrm{C}$ and $60^{\circ} \mathrm{C}$. Then all rods have equal length
(A) Thermal current in $A C$ and $C B$ are equal
(B) Thermal current in $A C$ and $A D$ are equal
(C) Temperature of $C$ and $D$ are unequal
(D) Thermal current in $A C$ and $B D$ is same

Hunza Gilgit

Numerade Educator

Problem 28

Assertion: Water can be boiled inside satellite by convection.
Reason: Convection is the process in which heat is transmitted from a place of higher temperature to a place of lower temperature by means of particles with their migrations from one place to another.
(A) A
(B) $\mathrm{B}$
(C) $\overline{\mathrm{C}}$
(D) D

Sunita Kumari

Numerade Educator

Problem 29

Assertion: If liquid in a container is heated at top rather than at the bottom. The main process by which the rest of the liquid becomes hot is conduction. Reason: Two spheres of the same material have radii $a$ and $2 a$, temperature $2 T$ and $T$, respectively. The energy radiated per second by the first sphere is greater than that of the second.
(A) $\mathrm{A}$
(B) $\mathrm{B}$
(C) $\mathrm{C}$
(D) D

Sunita Kumari

Numerade Educator

Problem 30

Heat energy is transferred from a heat source maintained at temperature $T_{1}=300^{\circ} \mathrm{C}$ to two lower temperature heat reservoirs maintained at temperatures $T_{2}=50^{\circ} \mathrm{C}$ and $T_{3}=100^{\circ} \mathrm{C}$. Three identical solid steel rods, each of length $L=1 \mathrm{~m}$ and cross-sectional area $0.01 \mathrm{~m}^{2}$, are used to pipe the heat, as shown. Find the ratio of power in watts delivered by the source to each of the lower temperature reservoirs and also find the temperature of the junction $O$. Consider only axial heat flow. Thermal conductivity of each of the rod is 46 SI units.

Narayan Hari

Numerade Educator

Problem 31

A liquid cools from $70^{\circ} \mathrm{C}$ to $60^{\circ} \mathrm{C}$ in $5 \mathrm{~min} .$ Calculate the time taken by the liquid to cool from $60^{\circ} \mathrm{C}$ to $50^{\circ} \mathrm{C}$, if the temperature of the surrounding is constant at $30^{\circ} \mathrm{C} .$

Sunita Kumari

Numerade Educator

Problem 32

Inner surface of a cylindrical shell of length $l$ and of material of thermal conductivity $k$ is kept at constant temperature $T_{1}$ and outer surface of the cylinder is kept at constant temperature $T_{2}$ such that $T_{1}>T_{2}$ as shown in Fig. 11.3. Heat flows from inner surface to outer surface radially outward. Inner and outer radii of the shell are $R$ and $2 R$, respectively. Due to lack of space, this cylinder has to be replaced by a smaller cylinder of length $\frac{l}{2}$, inner and outer radii $\frac{R}{4}$ and $R$, respectively, and thermal conductivity of material $n k$. If rate of radial outward heat flow remains same for same temperatures of inner and outer surface, i.e., $T_{1}$ and $T_{2}$, then find the value of $n$.

Sunita Kumari

Numerade Educator

Problem 33

Which of the following is more close to a black body?
(A) Black board paint
(B) Green leaves
(C) Black holes
(D) Red roses

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 34

Infrared radiations are detected by
(A) spectrometer
(B) pyrometer
(C) nanometer
(D) photometer

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 35

Two spheres of the same material have radii $1 \mathrm{~m}$ and $4 \mathrm{~m}$ and temperatures $4000 \mathrm{~K}$ and $2000 \mathrm{~K}$, respectively. The ratio of the energy radiated per second by the first sphere to that by the second is
(A) $1: 1$
(B) $16: 1$
(C) $4: 1$
(D) $1: 9$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 36

According to Newton's law of cooling, the rate of cooling of a body is proportional to $(\Delta \theta)^{n}$ where $\Delta \theta$ is the difference between the temperature of the body and the surrounding. Then $n$ is equal to
(A) 2
(B) 3
(C) 4
(D) 1

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 37

If the temperature of the sun was to increase from $T$ to $2 T$ and its radius from $R$ to $2 R$, then the ratio of the radiant energy received on earth to what it was previously will be
(A) 4
(B) 16
(C) 32
(D) 64

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 38

The temperatures of the two outer surfaces of a composite slab, consisting of two materials having coefficients of thermal conductivity $K$ and $2 K$ and thickness $x$ and $4 x$, respectively, are $T_{2}$ and $T_{1}$ $\left(T_{2}>T_{1}\right)$. The rate of heat transfer through the slab in a steady state is $\left[\frac{A\left(T_{2}-T_{1}\right) K}{x}\right] f$ with $f$ equal to
(A) 1
(B) $1 / 2$
(C) $2 / 3$
(D) $1 / 3$

Sunita Kumari

Numerade Educator

Problem 39

The Figure $11.4$ shows a system of two concentric spheres of radii $r_{1}$ and $r_{2}$ kept at temperature $T_{1}$ and $T_{2}$. The radial rate of flow of heat in a substance between the two concentric spheres is proportional to
(A) $\frac{\left(r_{2}-r_{1}\right)}{\left(r_{1} r_{2}\right)}$
(B) $\operatorname{In}\left(\frac{r_{2}}{r_{1}}\right)$
(C) $\frac{r_{1} r_{2}}{\left(r_{2}-r_{1}\right)}$
(D) $\left(r_{2}-r_{1}\right)$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 40

Assuming the sun to be a spherical body of radius $R$ at a temperature of $T \mathrm{~K}$. Evaluate the total radiant power, incident on earth, at a distance $r$ from the sun
(A) $\frac{4 \pi r_{0}^{2} R^{2} \sigma T^{4}}{r^{2}}$
(B) $\frac{\pi r_{0}^{2} R^{2} \sigma T^{4}}{r^{2}}$
(C) $\frac{r_{0}^{2} R^{2} \sigma T^{4}}{4 \pi r^{2}}$
(D) $\frac{R^{2} \sigma T^{4}}{r^{2}}$
where $r_{0}$ is the radius of the earth and $s$ is Stefan's constant.

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 41

One end of a thermally insulated rod is kept at a temperature $T_{1}$ and the other end at $T_{2}$. The rod is composed of two sections of length $l_{1}$ and $l_{2}$ and thermal conductivities $K_{1}$ and $K_{2}$, respectively. The temperature at the interface of the two sections is
(A) $\left(K_{2} l_{2} T_{1}+K_{1} l_{1} T_{2}\right) /\left(K_{1} l_{1}+K_{2} l_{2}\right)$
(B) $\left(K_{2} l_{1} T_{1}+K_{1} l_{2} T_{2}\right) /\left(K_{2} l_{1}+K_{1} l_{2}\right)$
(C) $\left(K_{1} l_{2} T_{1}+K_{2} l_{1} T_{2}\right) /\left(K_{1} l_{2}+K_{2} l_{1}\right)$
(D) $\left(K_{1} l_{1} T_{1}+K_{2} l_{2} T_{2}\right) /\left(K_{1} l_{1}+K_{2} l_{2}\right)$

RZ

Rubeena Zulfiqar

Numerade Educator

Problem 42

A liquid in a beaker has temperature $q(t)$ at time $t$, and if $\theta_{0}$ is the temperature of the surrounding, then according to Newton's law of cooling, the correct graph between $\log e\left(\theta-\theta_{0}\right)$ and $t$ is
(A)
(B)
(C)
(D)

Sunita Kumari

Numerade Educator

Problem 43

If a piece of metal is heated to temperature $\theta$ and then allowed to cool in a room, which is at temperature $\theta_{0}$, then the graph between the temperature $T$ of the metal and time $t$ will be closed to
(A)
(B)
(C)
(D)

Sunita Kumari

Numerade Educator