Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional fo

Question

Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional force are 10% of the input energy. How much power is generated by the turbine? (g=10 m/s²)

Accepted Answer

1. **Calculate Input Power ($P_{in}$):** The input power is the rate at which gravitational potential energy is supplied by the falling water. The rate of mass flow is $\frac{dm}{dt} = 15	ext{ kg/s}$.
   $$ P_{in} = \frac{d(mgh)}{dt} = \left(\frac{dm}{dt}ight)gh $$
   $$ P_{in} = (15	ext{ kg/s})(10	ext{ m/s}^2)(60	ext{ m}) = 9000	ext{ J/s} = 9000	ext{ W} = 9	ext{ kW} $$
2. **Calculate Output Power ($P_{out}$):** The problem states that 10% of the energy is lost to friction. Therefore, the turbine utilizes 90% of the input energy.
   $$ 	ext{Efficiency } (\eta) = 100\% - 10\% = 90\% = 0.9 $$
   $$ P_{out} = \eta 	imes P_{in} = 0.9 	imes 9	ext{ kW} = 8.1	ext{ kW} $$ [Class 11 Physics, Ch 6, Sec 6.10].

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