Question

A person decides to use his bath tub water to generate electric power to run a 40 watt bulb. The bath tub is located at a height of 10 m from the ground & it holds 200 litres of water. If we install a water driven wheel generator on the ground, at what rate should the water drain from the bath tub to light bulb? How long can we keep the bulb on, if the bath tub was full initially? The efficiency of generator is 90%. ($g=10m/s^{-2}$)

Answer 1

To power a 40 watt bulb, the generator should provide 40 J of energy per second.

Now if the generator efficiency is 90% then the input energy per second to generator should be-

$E_{in}=\frac{E_{out}}{0.9}$ or,

$E_{in}=\frac{40}{0.9}$ J or,

$E_{in}=\frac{400}{9}$ J

Now assume $m_r$ mass of water fall on wheel per second, then energy lost by water or energy transferred to wheel per second is

$E_{transferred}=m_r\times g\times h$

and we have

$E_{in}=E_{transferred}$

thus

$E_{in}=m_r\times 10\times 10$ or,

$\frac{400}{9}=m_r\times 10\times 10$ or,

$m_r=\frac{4}{9}$ Kg

Thus the mass fall rate is $\frac{4}{9}$ Kg/s.

Now, density of water is 1 Kg/L thus 200 L of water has mass

$m=200$ Kg

Now total time that 200 L of water power the 40 W bulb is

$t=\frac{m}{m_r}$ or,

$t=\frac{200}{4/9}$ s or,

$t=\frac{200\times 9}{4}$ s or,

$t=450$ s

thus total time the bulb can be kept on is 450 s.