Question

Water falls from a height of $60m$ at the rate of $15kg/s$ to operate a turbine. The losses due to frictional forces are $10\%$ of energy. How much power is generated by the turbine? $(g = 10m/{s^2})$

Answer 1

Hint Any falling object, or any object under the force of gravity will possess potential energy due to gravity, known as gravitational potential energy. The law of conservation of energy states that energy can neither be created nor be destroyed. It can only be converted from one form to another. Use these concepts to find the answer.

Complete step by step answer
Falling water will possess potential energy due to gravity. Let it be denoted by $PE$ . If $m$ is the mass of water that falls on the turbine and $h$ is the distance from the water to the turbine(measured from the turbine), we get
$PE = mgh$
We can now define the power or the rate of change of energy as
$P = \dfrac{{d(PE)}}{{dt}}$
$\Rightarrow P = gh\left( {\dfrac{m}{s}} \right)$
Substituting the value of the height and the amount of water that falls each second, we get
$P = 10 \times 60 \times 15$
$\Rightarrow P = 9000J/s$
The power can also be written in terms of watts $9000W$ . This means that the power generated by the falling water should be $9000W$ . It is given in the question that there is a power loss $10$ of the total power due to friction. So the total power generated after taking into consideration the power loss due to friction will be $90$ of the original power. So, we can compute the total power generated by the turbine as
$P = \dfrac{{d(PE)}}{{dt}} \times \dfrac{{90}}{{100}}$
By substituting the value of the original power, we get
$P = 9000 \times \dfrac{{90}}{{100}}$
$\Rightarrow P = 8100W = 8.1kW$

Hence, the total power generated by the turbine is found to be $8.1kW$ .

Note
Note that the water has only potential energy when it just begins to fall. While it is falling, the potential energy of the water gets converted to its kinetic energy and just when it reaches the turbine, the total potential energy would be converted to its kinetic energy and this kinetic energy is responsible for the generation of power.