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Water falls from a height of 60 m at the rate of 15 kg/s to operate a turbine. The losses due to frictional forces are 10% of energy. How much power is generated by the turbine? ($g=10m/{{s}^{2}}$)
A. 123 kW
B. 7 kW
C. 8.1 kW
D. 102 kW

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Last updated date: 13th Jun 2024
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Answer
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Hint: When an object is at height from the ground, it possesses gravitational potential energy proportional to its height. According to the law of conservation of energy, energy can neither be converted nor be destroyed. Therefore, the whole input energy except the losses is used to generate power. Power is defined as the rate of transfer of energy.

Formula used: Gravitational potential energy, $E=mgh$, $P=\dfrac{dE}{dt}$

Complete step by step answer:
When water is at height, it possesses gravitational potential energy. We have assumed that it does not possess any kinetic energy at this height.
Gravitational potential energy of water of mass m at height h is given by
$E=mgh$
Since water is falling, the rate of change of this energy is
$\dfrac{dE}{dt}=\dfrac{d}{dt}mgh$
Height h and acceleration due to gravity g is constant. Therefore,
$\dfrac{dE}{dt}=\dfrac{dm}{dt}gh=15\times 10\times 60=9000W$
According to the law of conservation of energy, energy can neither be converted nor be destroyed. Therefore, the whole input energy except the losses is used to generate power.
Water possesses 9kW of power at this height but due to frictional losses of 10%, only 90% input is used to generate power. Power is defined as the rate of transfer of energy. Therefore,
$P=\dfrac{dE}{dt}\times \dfrac{90}{100}=9kW\times 0.9=8.1kW$

So, the correct answer is “Option C”.

Note: If water has any kinetic energy at the height it is falling from, then that energy will also be used to generate power.
Also note that, to generate electricity in a turbine, water must possess kinetic energy. When water falls from height, its potential energy changes to kinetic energy and then this kinetic energy is used to rotate turbines and thus produce electricity.