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: In this question, the term frictional force is given which is a type of force that is generated between two surfaces that are in contact when slides against each other. Here given that some power is lost due to friction so we can find the power due to the remaining force that is occurring due to water falling on the turbine.
Formula used:
Power due to the work done per unit time by falling 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:
As given in the problem that water is falling from a height $(h)$ of $60m$at the rate of $15kg/s$. Hence
$h = 60m$
$m/t = 15kg/s$
During its fall the water does some work on the turbine to rotate the turbine when it falls on it. Here the work done by the water is given by
$w = mgh$.
This much work is done by the water to provide energy to the turbine to produce power $(P)$. As we know that power is the rate of doing work $(w)$ which is also the energy that is converted per unit of time. Hence
$P = \dfrac{w}{t}$
$ \Rightarrow P = \dfrac{{mgh}}{t}$
Here also given that the $10\% $power is lost due to frictional force hence the $90\% $ of the energy will be responsible for the power generation by the turbine. Hence the power generated ${P_{gen}}$is given by
${P_{gen}} = P \times 90\% $
$ \Rightarrow {P_{gen}} = \dfrac{{mgh}}{t} \times 90\% $ --------------- Equation $(1)$
Substituting the values of Rate of mass,$m/t = 15kg/s$ , height $h = 60m$, and $g = 10m/{s^2}$ in Equation $(1)$
${P_{gen}} = \dfrac{{15kg \times 60m \times 10m/{s^2}}}{{1s}} \times \dfrac{{90}}{{100}}$
$ \Rightarrow {P_{gen}} = 8100W$
$\therefore {P_{gen}} = 8.1kW$
Hence the power generated by the turbine by the remaining $90\% $ energy of the water falling from the height of $60m$ and at the rate of $15kg/s$ is ${P_{gen}} = 8.1kW$

Note: Power can be referred to as the ability of a body for doing work. It can also be defined as the amount of energy that is transferred per unit of time. Also, power is the rate of work done. Hence power can be defined in many ways so one must be aware of different quantities and ensure to use the correct terms and quantity of power in calculations.