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 →A semicircular rod is joined at its end to a straight rod of
Question: 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. Solution:...
Read More →Three rods of lengths 20cm each and area of cross - section
Question: 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 $\mathrm{K}_{\mathrm{AC}}=50 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}, \mathrm{K}_{\mathrm{BC}}=200 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}$ and $\mathrm{K}_{\mathrm{AC}}=400 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}$. Then junctions $\mathrm{A}, \mathrm{B}$ and $\mat...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sec ^{-1}\left(\frac{1}{1-2 x^{2}}\right)$ Solution:...
Read More →Consider the situation of the previous problem.
Question: 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 $1.0 \mathrm{~m}$. 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} / \...
Read More →On a winter day when the atmospheric temperature drops to
Question: 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 ...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\cot ^{-1}\left(\frac{\sqrt{1-x^{2}}}{x}\right)$ Solution:...
Read More →Figure shows water in a container having 2.0 mm
Question: Figure shows water in a container having $2.0 \mathrm{~mm}$ thick walls made of a material of thermal conductivity $0.50 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ 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 ste...
Read More →A cubical box of volume
Question: 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. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{x}{1+\sqrt{1-x^{2}}}\right)$ Solution: Ans) $\frac{1}{2 \sqrt{1-x^{2}}}$...
Read More →The ends of a metre stick are maintained at
Question: 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? Solution:...
Read More →The left end of a copper rod (length = 20cm, area of cross - section =0.20cm2)
Question: The left end of a copper rod (length $=20 \mathrm{~cm}$, area of cross - section $=0.20 \mathrm{~cm}^{2}$ ) 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 $=385 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$. Solution:...
Read More →Water at 50°C is filled in a closed cylindrical vessel of height 10cm
Question: 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 $1 \mathrm{~mm}$ thick Aluminium $\left(\mathrm{K}=200 \mathrm{~J} / \mathrm{m}-\mathrm{s}^{\circ} \mathrm{C}\right)$. Assume that the outside temperature is $20^{\circ} \mathrm{C}$. The density of water is $1000 \mathrm{~kg} / \mathrm{m}^{3}$, and the specific heat ca...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\tan ^{-1}\left(\frac{x}{\sqrt{1-x^{2}}}\right)$ Solution:...
Read More →A steel frame (K = 45W/m-°C) of total length 60cm and
Question: A steel frame $\left(\mathrm{K}=45 \mathrm{~W} / \mathrm{m}^{\circ} \mathrm{C}\right)$ 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. Solution:...
Read More →A pitcher with 1mm thick porous walls contains 10kg of water.
Question: 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 $0.1 \mathrm{~g} / \mathrm{s}$. The surface area of the pitcher (one side) $=200 \mathrm{~cm}^{2}$ The room temperature $=42^{\circ} \mathrm{C}$, latent heat of vaporization $=2.27 \times 10^{6} \mathrm{~J} / \mathrm{kg}$ and the thermal conductivity of the porous walls $=0.80 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}$ C...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sec ^{-1}\left(\frac{1}{\sqrt{1-x^{2}}}\right)$ Solution:...
Read More →An ice box almost completely filled with ice at 0°C
Question: An ice box 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 $2.0 \mathrm{~mm}$ and thermal conductivity $0.06 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C}$ Calculate the rate at which the ice melts in the box. Latent heat of fusion of ice $=3.4 \times 10^{5} \mathrm{~J} / \mathrm{kg}$ Solution:...
Read More →One end of a steel rod (K=46 J/m-s-°C.) of length of 1.0m
Question: One end of a steel rod $\left(\mathrm{K}=46 \mathrm{~J} / \mathrm{m}-\mathrm{s}-{ }^{\circ} \mathrm{C}\right.$. $)$ of length of $1.0 \mathrm{~m}$ 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 $=3.36 \times 10^{6} \mathrm{~J} / \mathrm{kg}$. ...
Read More →Water is boiled in a container having a bottom of surface area
Question: Water is boiled in a container having a bottom of surface area $225 \mathrm{~cm}^{2}$, thickness $1.0 \mathrm{~mm}$ and thermal conductivity $50 \mathrm{~W} / \mathrm{m}-{ }^{\circ} \mathrm{C} 100 \mathrm{~g}$ 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 $=2.26 \times 10^{6} \mathrm{~J} / \mathrm...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left(1-2 x^{2}\right)$ Solution:...
Read More →The normal body - temperature of a person
Question: 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 $=1.6 \mathrm{~m}^{2}$ conductivity of the cloth $=0.04 \mathrm{~J} / \mathrm{m}-\mathrm{s}^{\circ} \mathrm{C}$, thickness of the cloth $=0.5 \mathrm{~cm}$. Solution:...
Read More →A liquid-nitrogen container is made of a 1cm thick thermocoal
Question: A liquid-nitrogen container is made of a $1 \mathrm{~cm}$ thick thermocoal sheet having thermal conductivity $0.025$ $\mathrm{J} / \mathrm{m}-\mathrm{s}^{\circ} \mathrm{C}$ 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. Solution:...
Read More →A uniform slab of dimension 10cm×10cm×1cm
Question: 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 touches the reservoirs. The thermal conductivity of the material is $0.08$ $\mathrm{W} / \mathrm{m}^{\circ} \mathrm{C} \mathrm{C}$ Find the amount of heat flowing through the slab per minute. Solution:...
Read More →Differentiate each of the following w.r.t
Question: Differentiate each of the following w.r.t $x$ : $\sin ^{-1}\left(3 x-4 x^{3}\right)$ Solution:...
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