Question

Water falling from a $50$ m high fall is to be used for generating electric energy. If $1.8 \times {10^5}$ kg of water falls per hour and half the gravitational potential energy can be converted into electric energy, then how many $100$ W lamps can be lit?

Answer 1

Hint:Here, we have to calculate the number of lamps to be lit from the electrical energy produced by the waterfall. To calculate, we make use of the gravitational potential energy which is the energy associated with a body due to the virtue of its position above the surface of the earth.

Formula used:
If an object of mass $m$ is placed at a height $h$ above earth surface, then Gravitational potential energy is given as
$U = mgh$
Where, $g$- acceleration due to gravity of earth.

Complete step by step answer:
From the given data, we have $h = 50$ m , $m = 1.8 \times {10^5}$ kg per hour and $P = 100$ W.
Gravitational potential energy is given as
$U = mgh$
$\Rightarrow U = 1.8 \times {10^5} \times 9.8 \times 50$
This gives, $\therefore U = 882 \times {10^5}$ Joules per hour ------------(1)
Now, The half of this Gravitational potential energy is converted into electrical energy, thus
Electrical energy $ = \dfrac{1}{2} \times $ Gravitational potential energy
$E = \dfrac{1}{2} \times U = \dfrac{{882}}{2} \times {10^5}$
$\therefore E = 441 \times {10^5}$ Joules per hour ----------------(2)
We have to convert this electrical energy into power, because, lamp uses the electrical energy in terms of power (watts) , so
Electric power $ = \dfrac{{441 \times {{10}^5}}}{{3600}}$ Joules per second (Watts) --------------(3)
Number of $100$ watts lamps that can be lit from electric power $ = \dfrac{{441 \times {{10}^5}}}{{3600 \times 100}} = 122$

So, $122$ lamps of $100$ W can be lit from the electrical energy produced by the waterfall.

Note:We should convert the electrical energy in Joules per hour into Joules per seconds as we are asked to light lamps of watts and we know that \[1W = 1J{s^{ - 1}}\]. The value of acceleration due gravity of earth is a constant and it is $g = 9.8\,m{s^{ - 2}}$.