Relation between HORSEPOWER and TORQUE?

Here is one way to do it:

Mechanical work is defined as a force acting through a distance. In the foot, pound, second system measurement system, work is measured in foot-pounds.

Power is work done per period of time. That makes power foot-pounds per second.

One horsepower (Hp) is defined as 550 foot-pounds per second or 33,000 foot-pounds per minute.

Therefore: Hp = {[Force(lbs) X Distance(ft)] / Time(min} / 33,000

For something that is rotating, the distance / time part is (circumference X revolutions / minute).

Circumference is 2Pi X radius.

Torque is force X radius, so force = torque / radius.

Rather than force X distance / time you have: (torque / radius) X 2Pi X radius X RPM

Therefore: Hp = Torque(lbs-ft) X RPM X 2Pi / 33,000

Instead of multiplying by 2Pi and dividing by 33,000, divide by 33,000/ 2Pi = 5252.

You could also start with one watt = one newton-meter per second.

horsepower = 2 x pi x rpm x torque(lb.feet) / 33000