Thermal Resistance

Q.

Three rods of the same dimension have thermal conductivities 3K, 2K and K.They are arranged as shown in fig. Given below, with their ends at ${100}^{\circ }$ C, ${50}^{\circ }$ C and ${20}^{\circ }$ C. The temperature of their junction is

1. 35°C

2. 60°C

3. 70°C

4. 50°C

Q. A solid cylinder of radius $R$ made of a material of thermal conductivity $K_1$ is surrounded by a cylindrical shell of inner radius $R$ and outer radius $2R$ made of a material of thermal conductivity $K_2.$ The two ends of the combined system are maintained at two different temperatures. If there is no loss of heat across the cylindrical surface and the system is in steady state, the effective thermal conductivity of the system is:

1. $\frac{4K_1+3K_2}{4}$
2. $\frac{3K_1+4K_2}{4}$
3. $\frac{3K_1+K_2}{4}$
4. $\frac{K_1+3K_2}{4}$

Q. Two metal rods $1$ and $2$ of same lengths have the same temperature difference between their ends. Their thermal conductivities are $K_1$ and $K_2$, and cross-sectional areas $A_1$ and $A_2$ respectively. If the rate of heat conduction in $1$ is four times that in $2$, then

1. $K_1{A}_1=K_2{A}_2$
2. $K_1{A}_1=4K_2{A}_2$
3. $K_1{A}_1=2K_2{A}_2$
4. $4K_1{A}_1=K_2{A}_2$

Q. The two opposite faces of a cubical piece of iron (thermal conductivity $=0.2\text{CGS units}$) are at ${100}^{\circ }\text{C}$ and $0^{\circ }\text{C}$. The face at $0^{\circ }\text{C}$ is in contact with the ice at $0^{\circ }\text{C}$. If the area of a surface is $4{\text{cm}}^2$ and latent heat of fusion of ice is $80\text{J/g}$, then the mass of ice melted in $10$ minutes will be:

1. $30\text{gm}$
2. $300\text{gm}$
3. $5\text{gm}$
4. $50\text{gm}$

Q. Statement 1 : Equivalent thermal conductivity of two rods of same physical dimensions and having same thermal conductivity $\left(k\right)$ connected to each other in parallel combination is equal to $k$.
Statement 2 : Equivalent thermal conductivity of two rods of same physical dimensions and having same thermal conductivity $\left(k\right)$, connected to each other in series combination is equal to $k$.

1. Statement 1 is true but statement 2 is false.
2. Statement 2 is true but statement 1 is false.
3. Both statement 1 and statement 2 are true.
4. Both statement 1 and statement 2 are false.

Q. In the figure shown below, two metallic slabs of same area of cross section and thickness having thermal conductivities $500{\text{W/m}}^{\circ }\text{C}$ and $600{\text{W/m}}^{\circ }\text{C}$ respectively are connected to the same temperature difference in parallel combination. Find the equivalent thermal conductivity (in ${\text{W/m}}^{\circ }\text{C}$)

1. $600$
2. $500$
3. $550$
4. $450$

Q. A steam cylindrical pipe of inner radius $5\text{cm}$ and outer radius $7\text{cm}$, having conductivity $0.07\text{W/mK}$, carries steam at ${100}^{\circ }\text{C}$. If the temperature at the outer wall of the pipe jacket is ${20}^{\circ }\text{C}$, then find the heat lost through the jacket per metre length in an hour.

1. $1.28\times {10}^5\text{J}$
2. $9.72\times {10}^5\text{J}$
3. $3.76\times {10}^5\text{J}$
4. $7.84\times {10}^5\text{J}$

Q. A wall of thickness $0.6\text{m}$ has a normal area $1.5{\text{m}}^2$ and is made up of a material of thermal conductivity $0.4\text{W/mK}$. The temperature difference between the two sides is ${800}^{\circ }\text{C}.$. Find the thermal resistance of the wall.

1. $1\text{W/K}$
2. $1.8\text{W/K}$
3. $1\text{K/W}$
4. $1.8\text{K/W}$

Q. A wooden bowl conducts less heat than a copper bowl because

1. $k_{Wood}>k_{Copper}$
2. $k_{Wood}\le k_{Copper}$
3. $k_{Wood}=k_{Copper}$
4. $k_{Wood}<k_{Copper}$

Q. An electric heater is used in a room of total wall area $137{\text{m}}^2$ to maintain a temperature of ${20}^{\circ }\text{C}$ inside it when the outside temperature is $-{10}^{\circ }\text{C}.$ The walls have three layers of different materials. The innermost layer is of wood of thickness $2.5\text{cm}$, the middle layer is of cement of thickness $1.0\text{cm}$ and the outermost layer is of brick of thickness $25\text{cm}$. Find the power of the electic heater. Assume that there is no heat loss through the floor and ceiling. The thermal conductivities of wood, cement and brick are $0.125{\text{W/m}}^{\circ }\text{C}$, $1.5{\text{W/m}}^{\circ }\text{C}$ and $1.0{\text{W/m}}^{\circ }\text{C}$ respectively.

1. $9\text{W}$
2. $22.5\text{W}$
3. $9\text{kW}$
4. $22.5\text{kW}$

Q. Three rods of same dimensions have thermal conductivities $3\text{K}$, $\text{2K}$ and $\text{K}$. They are arranged as shown in the figure, with their ends at ${100}^{\circ }\text{C},{50}^{\circ }\text{C}$ and $0^{\circ }\text{C}.$ The temperature of their junction point is:

1. ${75}^{\circ }\text{C}$
2. ${\frac{200}{3}}^{\circ }\text{C}$
3. ${40}^{\circ }\text{C}$
4. ${\frac{100}{3}}^{\circ }\text{C}$

Q. Two rods of same length, cross sectional area and material transfer a given amount of heat in $12$ seconds, when they are joined in parallel. If they are joined in series with same temperature difference across the ends, then they will transfer same heat in

1. $24\text{s}$
2. $3\text{s}$
3. $1.5\text{s}$
4. $48\text{s}$

Q. Two slabs each of area $A$, length $2\text{m}$ and $4\text{m}$, thermal conductivities $k_1=300{\text{W/m}}^{\circ }\text{C}$ and $k_2=200{\text{W/m}}^{\circ }\text{C}$ respectively are joined to form a single slab of length $6\text{m}$. Find the equivalent thermal conductivity of resultant slab. (in ${\text{W/m}}^{\circ }\text{C}$)

1. $300$
2. $225$
3. $325$
4. $275$

Q. Two rods of different materials and identical cross sectional area, are joined face to face at one end and their free ends are fixed to the rigid walls. If the temperature of the surroundings is increased by ${30}^{o}C$, the magnitude of the displacement of the joint of the rod is (length of rods $l_1=l_2=1$ unit, ratio of their young's moduli $Y_1/Y_2=2$, coefficient of linear expansion are ${\alpha }_1$ and ${\alpha }_2$)

1. $5({\alpha }_2-{\alpha }_1)$
2. $10({\alpha }_1-{\alpha }_{21})$
3. $10({\alpha }_2-{2\alpha }_1)$
4. $5({2\alpha }_1-{\alpha }_2)$

Q. A composite block is made of slabs $A,B,C,D$ and $E$ of different thermal conductivities (given in terms of a constant $K$) and sizes (given in terms of length $L$ and area of cross-section $S$) as shown in the figure. All slabs are of the same width. Heat $Q$ flows only from left to right through the blocks. Then in the steady state:

1. heat flow through slabs $A$ and $E$ are same
2. heat flow through slab $B$ is minimum
3. temperature difference across slab $E$ is smallest
4. heat flow through $C=$ heat flow through $B+$ heat flow through $D$

Q. Three identical rods of length $1\text{m}$ each, having cross-sectional area of $1{\text{cm}}^2$ each and made of aluminium, copper and steel respectively are maintained at temperatures of ${12}^{\circ }\text{C},4^{\circ }\text{C}$ and ${50}^{\circ }\text{C}$ respectively at their separate ends. Find the temperature of their common junction.
$[K_{Cu}=400\text{W/m-K},K_{Al}=200\text{W/m-K},K_{\text{Steel}}=50\text{W/m-K}]$

1. ${10}^{\circ }\text{C}$
2. ${20}^{\circ }\text{C}$
3. ${30}^{\circ }\text{C}$
4. ${40}^{\circ }\text{C}$

Q. Two metallic slabs of same area of cross section and length having thermal conductivities $300{\text{W/m}}^{\circ }\text{C}$ and $200{\text{W/m}}^{\circ }\text{C}$ respectively are connected in series combination. Find the equivalent thermal conductivity (in ${\text{W/m}}^{\circ }\text{C}$)

1. $240$
2. $360$
3. $250$
4. $275$

Q.

Three rods of identical area of cross-section and

made from the same metal form the sides of an

isosceles triangle ABC , right angled at B. The

points A and B are maintained at temperatures

T and $\sqrt{2}$T respectively. In the steady state the

temperature of the point C is $T_c$. Assuming that

only heat conduction takes place, $\frac{T_C}{T}$ is equal to

1. 

2. 

3. 

4. 

Q. The two opposite faces of a cubical piece of iron (thermal conductivity $=0.2\text{CGS units}$) are at ${100}^{\circ }\text{C}$ and $0^{\circ }\text{C}$. The face at $0^{\circ }\text{C}$ is in contact with the ice at $0^{\circ }\text{C}$. If the area of a surface is $4{\text{cm}}^2$ and latent heat of fusion of ice is $80\text{J/g}$, then the mass of ice melted in $10$ minutes will be:

1. $30\text{gm}$
2. $300\text{gm}$
3. $5\text{gm}$
4. $50\text{gm}$

Q. For the figure shown below, choose the option that correctly represents the combination.

1. $K_1$ and $K_2$ are in series combination and together they are in parallel combination with $K_3$.
2. $K_1$ and $K_2$ are in parallel combination and together they are in series combination with $K_3$.
3. $K_1,K_2$ and $K_3$ all are in parallel combination with each other.
4. $K_1,K_2$ and $K_3$ all are in series combination with each other.

Q. Thermal conductivity ($k$) depends on the material, and not the dimensions and mass of the body.

1. False
2. True

Q. In the first experiment, two identical conducting rods are joined one after the other and this combination is connected to two vessels, one containing water at ${100}^{\circ }\text{C}$ and the other containing ice at $0^{\circ }\text{C}$ (see Fig). In the second experiment, the two rods are placed one on top of the other and ends are connected to the same vessels. If $q_1$ and $q_2$ (in gram per second) are the respective rates of melting of ice in the two cases, then the ratio $\frac{q_1}{q_2}$ is

1. $\frac{1}{2}$
2. $\frac{2}{1}$
3. $\frac{1}{4}$
4. $\frac{1}{8}$

Q. Equal temperature difference exists between the ends of two metallic rods $1$ and $2$ of length. Their thermal conductivities are $K_1$ and $K_2$ and cross sectional areas represents $A_1$ and $A_2$. The condition for equal rate of heat transfer is:

1. $K_1{A}_2=K_2{A}_1$
2. $K_1{A}_1=K_2{A}_2$
3. $K_1{A}_1^2=K_2{A}_2^2$
4. $K_1^2{A}_2=K_2^2{A}_1$

Q. A spherical body of radius $4\text{m}$ has a concentric cavity of radius $2\text{m}$ as shown in the figure. Thermal conductivity of the material is $\frac{100}{\pi }{\text{W/m}}^{\circ }\text{C}$. If the temperature of inner surface is kept at ${120}^{\circ }\text{C}$ and of the outer surface at ${40}^{\circ }\text{C}$. Then, find the temperature $T$ at a distance $3\text{m}$ from centre. (in ${}^{\circ }\text{C}$).

1. ${66.66}^{\circ }\text{C}$
2. ${33.66}^{\circ }\text{C}$
3. ${23}^{\circ }\text{C}$
4. ${63.32}^{\circ }\text{C}$

Q. The three rods shown in figure have identical dimensions. Heat flows from the hot end at a rate of $40\text{W}$ in the arrangement (a). Find the rates of heat flow when the rods are joined as in arrangement (b). (Assume $k_{Al}=200\text{W/m}{}^{\circ }\text{C}$ and $k_{Cu}=400\text{W/m}{}^{\circ }\text{C}$)

1. $75\text{W}$
2. $200\text{W}$
3. $400\text{W}$
4. $4\text{W}$

Q. The coefficient of thermal conductivity of copper is nine times that of steel. In the composite cylindrical bar shown in figure, what will be the temperature at the junction of copper and steel?

1. ${75}^{\circ }\text{C}$
2. ${67}^{\circ }\text{C}$
3. ${33}^{\circ }\text{C}$
4. ${25}^{\circ }\text{C}$

Q. Two ends of a conducting rod of varying cross-section are maintained at ${200}^{\circ }\text{C}$ and $0^{\circ }\text{C}$ respectively. In staedy state

1. $\text{temperature difference across}AB\text{and}CD\text{are equal}$
2. $\text{temperature difference across}AB\text{is greater than that of across}CD$
3. $\text{temperature difference across}AB\text{is less than that of across}CD$
4. $\text{temperature difference may be equal or different depending on the thermal conductivity of the rod}$

Q. 9. What is specific heat ?

Q. The container $A$ contains ice at $0^{\circ }\text{C}$. A conducting uniform rod $PQ$ of length $4R$ is used to transfer heat to the ice in the container. The end $P$ of the rod is maintained at ${100}^{\circ }\text{C}$ and the other end $Q$ is kept inside the container $A$. The complete ice melts in $23\text{minutes}$. In another experiments, two conductors in the shape of quarter circles of radii $2R$ and $R$ are welded to the conductor $PQ$ at $M$ and $N$ respectively and their other ends are inserted inside the container $A$. All conductors are made of the same material and have the same cross sectional area. Once again, the end $P$ is maintained at ${100}^{\circ }\text{C}$ and this time, the complete ice melts in $t\text{minutes}$. Find $t$.
[Assume no heat loss from the curved surface of the rods and take $\pi \simeq 3.0$]

1. $35\text{min}$
2. $21\text{min}$
3. $17.5\text{min}$
4. $12\text{min}$

Q. A spherical body of radius $b=\frac{4}{\pi }\text{m}$ has a concentric cavity of radius $a=\frac{2}{\pi }\text{m}$ as shown. Thermal conductivity of the material is $100{\text{W/m}}^{\circ }\text{C}$. Find the thermal resistance between inner and outer surface.

1. ${6.25\times {10}^{-4}}^{\circ }\text{C/W}$
2. ${1.25\times {10}^{-4}}^{\circ }\text{C/W}$
3. ${6.5\times {10}^{-4}}^{\circ }\text{C/W}$
4. ${3.75\times {10}^{-4}}^{\circ }\text{C/W}$