a) Show that one British horsepower is equal to 746 W. (b) What is the horsepower rating of a 75-W light bulb?

Power, P is defined as the rate at which work is done and is obtained by dividing the time taken to perform that work.

\(P = \frac{{{\rm{Work}}}}{{{\rm{Time}}}} = \frac{W}{t}\)

Units of power:

The SI unit is Watt (W) and the imperial unit is horsepower (hp).

Horsepower is a unit of power that compares the power of a machine to the horse muscle power. The power of turbines, heavy engines, and other machinery is usually measured in horsepower.

One horsepower is equal to\(550\;{\rm{ft - lb/s}}\).

Also,

\(1\;{\rm{lb}} = 4.45\;{\rm{N}}\)

\(1\;{\rm{m}} = 3.38\;{\rm{ft}}\)

Therefore, one hp can be converted into Watts as:

\(\begin{aligned}1\;{\rm{hp}} &= \left( {\frac{{550\;{\rm{ft - lb/s}}}}{{1\;{\rm{hp}}}}} \right)\left( {\frac{{4.45\;{\rm{N}}}}{{1\;{\rm{lb}}}}} \right)\left( {\frac{{1\;{\rm{m}}}}{{3.28\;{\rm{ft}}}}} \right)\\ &\approx 746\;{\rm{W}}\end{aligned}\)

Thus, one horsepower is equal to 746 Watts.

It is given that the input power of the bulb is 75 W. The rating of the bulb in hp is:

\(\begin{aligned}P' &= \left( {75\;{\rm{W}}} \right)\left( {\frac{{1\;{\rm{hp}}}}{{746\;{\rm{W}}}}} \right)\\ &= 0.10\;{\rm{hp}}\end{aligned}\)

Thus, the rating of the light bulb in terms of horsepower is 0.10 hp.