Question

Water from a stream is falling on the blades of a turbine at the rate of $100kg/s$ . If the height of the stream is $100m$ , then the power delivered to the turbine is
(A) $100kW$
(B) $0.1kW$
(C) $10kW$
(D) $1kW$

Answer 1

When water falls from a height, it has potential energy due to the force of gravity. This potential energy will be the energy that gets transferred to the turbine, which then produces power. Use this idea to proceed with the problem.

Complete step by step solution:
Let the mass of the water that falls be denoted by $m$ . Let the height from which the water falls as measured from the turbine be denoted by $h$ . Let the potential energy of the water that falls be denoted by $E$ .
The potential energy can be computed as follows
$E = mgh$
This energy is the same energy that gets converted into kinetic energy just when it touches the blades of the turbine. This kinetic energy is used to turn the turbine and hence generate power. So the rate of change of potential energy of the water will be equal to the power generated by the turbine due to the falling water.
Therefore,
$P = \dfrac{{dE}}{{dt}}$
Here $P$ is the total power generated.
Substituting the value of the potential energy into the above equation, we get
$P = \dfrac{{d(mgh)}}{{dt}}$
Substitute the value for the rate at which the water falls to the turbine $\left( {\dfrac{{dm}}{{dt}}} \right)$ and substitute the value for the height. This gives us
$P = 10 \times 100 \times 100$
$\Rightarrow P = 10000W = 10kW$
That is, the total power that can be obtained from the turbine is $10kW$ .

Hence, we can conclude the option (C) to be the correct option.

Note:
We have used the law of conservation of energy which states that energy can neither be created nor be destroyed. We have used the approximated value of acceleration due to gravity in this case. We have used $10m/{s^2}$ instead of $9.8m/{s^2}$ .