In Fig. 26-37, a resistance coil, wired to an external battery, is placed inside a thermally insulated cylinder fitted with a frictionless piston and containing an ideal gas. A current i = 240mA flows through the coil, which has a resistance. At what speed vmust the piston, of mass m=12kg , move upward in

Order that the temperature of the gas remains unchanged?

1. Current flowing through the coil, i = 240mA
2. Resistance of the coil, R = 550$\mathrm{\Omega }$
3. Mass of the piston, m = 12kg

As the piston moves up, its mechanical energy would increase. The rate at which the heat is supplied by the resistance coil should be equal to the rate of increase in the mechanical energy of the piston. By writing the equations for these two processes and equating them, we can find the velocity of the piston in the upward direction.

Formulae:

The electric power due to the current flow, $P=i^{2}R$ (i)

The rate of energy transfer due to speed of a body, $P=mgv$ (ii)

The rate of increase in the mechanical energy of the piston should bethesame as the rate of supply of thermal energy by the coil. Thus, equating equations (i) and (ii), we get the value of the speed of the piston by using the given data as follows:

$$\begin{gathered} \mathrm{mgv}={\mathrm{i}}^2\mathrm{R} \\ \mathrm{v}=\frac{{\mathrm{i}}^2\mathrm{R}}{\mathrm{mg}} \\ =\frac{{\left(0.240\right)}^2\left(550\right)}{\left(12\right)\left(9.8\right)} \\ =0.26939\mathrm{m}/\mathrm{s} \\ \approx 0.27\mathrm{m}/\mathrm{s} \end{gathered}$$

Hence, the value of the speed is 0.27 m/s .