You asked....

OK, if HP = RPM x TORQUE ÷ 5252, then where does the 5252 come from?

By definition, POWER = FORCE x DISTANCE ÷ TIME

Using an example above (not displayed here on PF), where a constant tangential force of 100 pounds was applied to the 12" handle rotating at 2000 RPM, we know the force involved, so to calculate power, we need the distance the handle travels per unit time, expressed as:

Power = 100 pounds x distance per minute

OK, how far does the crank handle move in one minute? First, determine the distance it moves in one revolution:

Distance per revolution = 2 x π x radius

Distance per revolution. = 2 x 3.1416 x 1 ft = 6.283 ft.

Now we know how far the crank moves in one revolution. How far does the crank move in one minute?

Distance per min. = 6.283 ft .per rev. x 2000 rev. per min. = 12,566 feet per minute

Now we know enough to calculate the power, defined as:

POWER = FORCE x DISTANCE ÷ TIME

so

Power = 100 lb x 12,566 ft. per minute = 1,256,600 ft-lb per minute

Great, but how about horsepower? Remember that one horsepower is defined as 33000 foot-pounds of work per minute. Therefore HP = POWER (ft-lb per min) ÷ 33,000. We have already calculated that the power being applied to the crank-wheel above is 1,256,600 ft-lb per minute.

How many HP is that?

HP = (1,256,600 ÷ 33,000) = 38.1 HP.

Now we combine some stuff we already know to produce the magic 5252. We already know that:

TORQUE = FORCE x RADIUS.

If we divide both sides of that equation by RADIUS, we get:

(a) FORCE = TORQUE ÷ RADIUS

Now, if distance per revolution = RADIUS x 2 x π, then

(b) distance per minute = RADIUS x 2 x π x RPM

We already know

(c) POWER = FORCE x DISTANCE per minute

So if you substitute the equivalent for force from equation (a) and distance per minute from equation (b) into equation (c), we get:

POWER = (TORQUE ÷ RADIUS) x (RPM x RADIUS x 2 x π)

Dividing both sides by 33,000 to find HP:

HP = TORQUE ÷ RADIUS x RPM x RADIUS x 2 x π ÷ 33,000

By reducing, we get

HP = TORQUE x RPM x 6.28 ÷ 33,000

Since

33,000 ÷ 6.2832 = 5252

Therefore

HP = TORQUE x RPM ÷ 5252

Note that at 5252 RPM, torque and HP are equal. At any RPM below 5252, the value of torque is always greater than the value of HP; Above 5252 RPM, the value of torque is always less than the value of HP.

I hope that makes some sense.