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.

Let us break down the solution step-by-step to understand how the power generated by the turbine is calculated.

Step 1: Identify the given information
- Height from which water falls ($h$) = 60 m
- Rate of flow of mass of water ($\frac{dm}{dt}$) = 15 kg/s
- Acceleration due to gravity ($g$) = 10 m/s²
- Frictional energy loss = 10% of the input energy

Step 2: Calculate the input power (rate of input gravitational potential energy)
The potential energy ($E_p$) of water stored at a height $h$ is given by the formula:
$E_p=m·g·h$

Since the water is falling continuously, the input power ($P_{\text{input}}$), which is the potential energy per unit time, can be expressed as:
$P_{\text{input}}=\frac{dm}{dt}·g·h$

Substituting the given values into the formula:
$P_{\text{input}}=15\text{kg/s}·10{\text{m/s}}^2·60\text{m}$

Simplifying the multiplication:
$P_{\text{input}}=9000\text{W}=9\text{kW}$

Step 3: Account for frictional losses
The losses due to frictional forces are 10% of the input energy (or input power). Therefore, the efficiency ($\eta$) of the turbine system is:
$\eta =100\%-10\%=90\%=0.9$

Step 4: Calculate the useful power generated by the turbine
The power generated ($P_{\text{output}}$) is the remaining 90% of the input power:
$P_{\text{output}}=\eta ·P_{\text{input}}$

Substituting the values:
$P_{\text{output}}=0.9·9\text{kW}$

Calculating the final product:
$P_{\text{output}}=8.1\text{kW}$

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