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

In this problem, we have provided that the stream of water is falling on the blades of the turbine which in turn provides the force to rotate the blades. Here we have to find the power delivered by the water to the turbine to do work. Power is a rate of doing work or the amount of energy transferred per unit of time.
Formula used:
Power due to the work done by the water per unit time
$P = \dfrac{{mgh}}{t}$
where $m =$ Mass of water
$g =$Gravitational acceleration
$h =$Height of waterfall

Complete step by step answer:
Here as the waterfalls on the blades of the turbine, it does work on the turbine's blade which in turn starts rotating. We have given that the water from the stream is falling at the rate of $100kg/s$ from a height of $100m$. As we have discussed that work done by the water per unit time is known as the power $P$ which can be given as
$P = \dfrac{{mgh}}{t}$ ------------------- Equation $(1)$
where $m =$ Mass of water
$g =$Gravitational acceleration
$h =$Height of waterfall
Now substituting all the values of rate of mass $m/t = 100kg/s$and the height $h = 100m$and $g = 10m/{s^2}$ in the equation $(1)$we get
$P = \dfrac{{100kg \times 10m/{s^2} \times 100m}}{{1s}}$
Calculating the above terms and multiplying them we get
$P = 100000W$
$\therefore P = 100kW$
So the power delivered by the water falling from the stream at the rate of $100kg/s$ from the height of $100m$ is given by $100kW$.

Hence option (1) is the correct answer.

Note: Here we have taken the value of gravitational acceleration $g = 10m/{s^2}$ which is an approximate value of $9.8m/{s^2}$. We can also use this value in our calculation but there will be and only a very slight difference will be observed so if the value $g$ is not mentioned in the problem then we can take any value either $9.8m/{s^2}$ or $10m/{s^2}$ according to our convenience.