Figure shows a paddle wheel coupled with an agitator submerged in a water tank placed in an ice bath. The thickness of tank walls is 2mm and with thermal conductivity $0 .5 W / m^{\circ} C$ . The surface area of tank in contact with water is $0 .05 m^{2}$ . As the block of mass M attached to wheel goes down agitator rotates. It is found that in steady state block goes down at a constant speed of 0.1m/s and water temperature remain constant at 1°C. Find the mass of block (in kg). $(Take g = 10 m / s^{2})$

Moderate

Solution

rate of loss in gravitational potential energy of block is equal to the rate of heat conduction from water to ice bath as T_water=constant.
rate of loss of gravitational P.E. is = Mg v
rate of the thermal conduction $\frac{dθ}{dt} = \frac{kA (T_{2} - T_{1})}{x} = \frac{0 .5 \times 0 .05 \times 1}{2 \times 10^{- 3}} = 12 .5 \frac{J}{s} .$
for steady temperature of water $Mg v = 12 .5$
$M = \frac{12 .5}{10 \times 0 .1} = 12 .5 kg$

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