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Question
Following Figure shows water in a container having 2.0 mm thick walls made of a material of thermal conductivity 0.50 W m−1°C−1. The container is kept in a melting-ice bath at 0°C. The total surface area in contact with water is 0.05 m2. 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 cm s−1 and the temperature of the water remains constant at 1.0°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 m s−2.
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Solution
Temperature of water, T1 = 1°C
Temperature if ice bath, T2 = 0°C
Thermal conductivity, K = 0.5 W/m °C
Length through which heat is lost, l = 2 mm = 2 × 10–3 m
Area of cross section, A = 5 × 10−2 m2
Velocity of the block, v = 10 cm/sec = 0.1 m/s
Let the mass of the block be m.
Power = F · v
= (mg) v ......(1)
Also,
`Power = (DeltaQ)/(Deltat) ..................(2)`
`(DeltaQ)/(Deltat) = (k.A ( T_1 - T_2 ))/l ................(3)`
From equation (1), (2) and (3), we get
`(mg)v = (k.A ( T_1 - T_2 ))/l `
`m =(0.5xx 5xx140^-2(1))/ ((2xx10^-3)xx10xx0.1`
`m = 12.5 kg`
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