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)

The correct option is 8.1 kW.

Step-by-step Explanation:

1. Understand the input parameters:
- Height from which water falls, $h=60\text{m}$
- Mass flow rate of water, $\frac{dm}{dt}=15\text{kg/s}$
- Acceleration due to gravity, $g=10{\text{m/s}}^2$
- Frictional energy loss = $10\%$ of the input energy.

2. Calculate the input power (rate of potential energy input):
The potential energy per unit time (input power $P_{\text{in}}$) of the falling water is given by:
$P_{\text{in}}=\frac{dE}{dt}=\frac{dm}{dt}\cdot g\cdot h$

Substituting the values:
$P_{\text{in}}=15\text{kg/s}\times 10{\text{m/s}}^2\times 60\text{m}$
$P_{\text{in}}=9000\text{W}=9.0\text{kW}$

3. Account for the frictional losses:
Since $10\%$ of the input energy is lost due to friction, the efficiency ($\eta$) of the turbine system is:
$\eta =100\%-10\%=90\%=0.9$

4. Calculate the output power generated by the turbine:
$P_{\text{out}}=\eta \times P_{\text{in}}$
$P_{\text{out}}=0.9\times 9.0\text{kW}$
$P_{\text{out}}=8.1\text{kW}$

Thus, the power generated by the turbine is 8.1 kW.