Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left\{2 a x \sqrt{1-a^{2} x^{2}}\right\}$ Solution:...
Read More →Seven rods A, B, C, D, E, F and G are joined as shown in figure.
Question: Seven rods A, B, C, D, E, F and $\mathrm{G}$ are joined as shown in figure. All the rods have equal cross - sectional area A and the length I. The thermal conductivities of the rods are $\mathrm{K}_{\mathrm{A}}=\mathrm{K}_{\mathrm{C}}=\mathrm{K}_{0}, \mathrm{~K}_{\mathrm{B}}=\mathrm{K}_{\mathrm{D}}=2 \mathrm{~K}_{0}, \mathrm{~K}_{\mathrm{E}}=3 \mathrm{~K}_{0}, \mathrm{~K}_{\mathrm{F}}=4 \mathrm{~K}_{0}$ and $\mathrm{K}_{\mathrm{G}}=5 \mathrm{~K}_{0}$. The rod $\mathrm{E}$ is kept at a ...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left\{\frac{x}{\sqrt{a^{2}-x^{2}}}\right\}$ Solution: ....
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cos ^{-1}\left(\frac{1-x^{2 n}}{1+x^{2 n}}\right)$ Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sec ^{-1}\left(\frac{x^{2}+1}{x^{2}-1}\right)$ Solution:...
Read More →Four identical rods AB, CD, CF and DE are joined as shown in figure.
Question: Four identical rods $\mathrm{AB}, \mathrm{CD}, \mathrm{CF}$ and $\mathrm{DE}$ are joined as shown in figure. The length, cross - sectional area and thermal conductivity of each rod are $I, A$ 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. Solution:...
Read More →The three rods shown in figure have identical geometrical dimensions.
Question: The three rods shown in 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 $200 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ and $400 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ respectively. Solution:...
Read More →The two rods shown in figure have identical geometrical dimensions.
Question: The two rods shown in figure 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. Solution:...
Read More →A room has a window fitted with a single 1.0m × 2.0m glass of
Question: A room has a window fitted with a 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 $32^{\circ} \mathrm{C}$ 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 o...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left(\frac{1}{\sqrt{1+x^{2}}}\right)$ Solution:...
Read More →Suppose the bent part of the frame of the previous problem
Question: Suppose the bent part of the frame of the previous problem has a thermal conductivity of $780 \mathrm{~J} / \mathrm{m}-\mathrm{s}^{\circ} \mathrm{C}$ whereas it is $390 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}$ 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. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sec ^{-1}\left(\frac{1+x^{2}}{1-x^{2}}\right)$ Solution:...
Read More →Consider the situation shown in figure.
Question: 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 $100^{\circ} \mathrm{C}$ is $130 \mathrm{~J}$. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\operatorname{cosec}^{-1}\left(\frac{1+x^{2}}{2 x}\right)$ Solution:...
Read More →Figure shows an aluminium rod joined to a copper rod.
Question: Figure 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 $0.20 \mathrm{~cm}^{2}$. 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 $K_{A l}=200 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ ...
Read More →An aluminium rod and a copper rod of equal length
Question: 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 figure. 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 $=200 \mathrm{~W} / \mathrm{m}$ ${ }^{\circ} \mathrm{C}$ and of copper $=390 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$. Sol...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{3 x-x^{3}}{1-3 x^{2}}\right)$ Solution:...
Read More →Figure shows a copper rod joined to a steel rod.
Question: 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 $0^{\circ} \mathrm{C}$ 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 $=390 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ and that of steel $=46 \mathrm{~W} / \mathrm{m}$ - ${ }^{\circ} \mathrm{C}$. Solution:...
Read More →A composite slab is prepared by pasting two plates
Question: A composite slab is prepared by pasting two plates of thicknesses $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ and thermal conductivity $\mathrm{K}_{1}$ and $\mathrm{K}_{2}$. The slabs have equal cross - sectional area. Find the equivalent conductivity of the slab. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\frac{1+x}{1-x}\right)$ Solution:...
Read More →A hollow tube has a length L,
Question: A hollow tube has a length $L$, inner radius $R_{1}$ and the outer radius $R_{2}$. The material has a thermal conductivity $\mathrm{K}$. Find the heat flowing through the walls of the tube if (a) the flat ends are maintained at temperature $T_{1}$ and $T_{2}\left(T_{1}T_{2}\right)$ and the inside of the tube is maintained at temperature $T_{1}$ and the outside is maintained at $\mathrm{T}_{2}$. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{1+x}{1-x}\right)$ Solution:...
Read More →A hole of radius r1 is made centrally in a uniform circular disc of thickness d and
Question: A hole of radius $r_{1}$ is made centrally in a uniform circular disc of thickness $d$ and radius $r_{2}$. The inner surface (a cylinder of length $d$ and radius $r_{1}$ ) is maintained at a temperature $\theta_{1}$ and the outer surface (a cylinder of length $d$ and radius $\left.r_{2}\right)$ is maintained at a temperature $\theta_{2}\left(\theta_{1}\theta_{2}\right)$. The thermal conductivity of the material of the disc is $\mathrm{K}$. Calculate the heat flowing per unit time throu...
Read More →Steam at 120°C is continuously passed through a 50cm
Question: Steam at $120^{\circ} \mathrm{C}$ is continuously passed through a $50 \mathrm{~cm}$ long rubber tube of inner and outer radii $1.0 \mathrm{~cm}$ 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 $=0.15 \mathrm{~J} / \mathrm{m}-\mathrm{s}^{\circ} \mathrm{C}$. Solution:...
Read More →A metal rod of cross - sectional area
Question: A metal rod of cross - sectional area $10 \mathrm{~cm}^{2}$ is being heated at one end. At one time, the temperature gradient is $5.0^{\circ} \mathrm{C} / \mathrm{cm}$ at cross - section $\mathrm{A}$ and is $2.5^{\circ} \mathrm{C} / \mathrm{cm}$ at cross-section $\mathrm{B}$. Calculate the rate at which the temperature is increasing in the part $\mathrm{AB}=0.40 \mathrm{~J} /{ }^{\circ} \mathrm{C}$, thermal conductivity of the material of the rod = $200 \mathrm{~W} / \mathrm{m}^{\circ}...
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