Figure shows water in a container having `2.0mm` thick walls made of a material of thermal conductivity `0.50Wm^(-1).^(@)C^(-1)` . The container is kept in a melting-ice bath at `0^(@)C` . The total surface area in contact with water is `0.05m^(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 steady state is reached in which the block goes down with a constant speed of `10cm^(-1)` and the temperature of the water remains constant at `1.0^(@)C` .Find the mass `M` of the block. Assume that the heat flow out of the water only through the walls in contact. Take `g=10ms^(-2)` .

Explanation

At steady state, rate of doing work by gravity on block M `=`Rate of energy produced
or `(d)/(dt)[Mgh]=(d)/(dt)[Q+(1)/(2)Mv^2]`
or `Mg((dh)/(dt))=(dQ)/(dt)+(M)/(2)xx2v(dv)/(dt)` ..(i)
As block goes down with constant velocity ,
`(dh)/(dt)=v` and `(dv)/(dt)=0`
`:. Mgv=(dQ)/(dt)`
Here `(dQ)/(dt)=KA(DeltaT)/(L)`
`:. Mgv=KA(DeltaT)/(L)`
or `Mxx10xx(0.1)=0.50xx0.50((1.0-0))/(2xx10^3)`
or `M=12.5kg`