(ft * lb * 2 * pi * radians) / (minute * hp)

Remember that one revolution (the diameter) is 2 * pi * radians? Now you can see why you don't want to go to radians. ;-) So how did that constant get such "ugly" units? The "problem" is Watt was simply observing that the horse lifted the 150 lb. object by 220 feet in one minute. We can't change that. With the definition of lb., pi (3.14159 etc.) , and feet already existing and defining one horsepower by what he observed (which comes to lifting 550 lb. by one foot in one second as the commonly listed definition of HP) he simply put the two actions (torque and HP) into an equation and derived whatever constant related the two. Since the two actions are "equivalent", the units of the constant are whatever makes both sides have the same units. This brings up the topic of "unit analysis." In the above Watt was deriving a basic relationship, but when we are trying to calculate something complicated we often use "unit analysis" as a check. That is we just look at the units for all the data (observed or given to us) we will plug into an equation and set up an "equation" with just the units on both sides and make sure they are equivalent. It is interesting to note that had Watt simply observed a bigger (say, a Clydesdale) or smaller horse (like Ford used to measure the '99 Cobra!) the definition of HP would be different. I get to make cracks like that since I am a Ford fan and own a '98 Cobra and my wife has a '95 Lightning.

Regards,

Steve

Here is a comment from Keith Menges

I came across the 'Horsepower and Torque' page while looking for the magic conversion number for horsepower (5252). Great page. It is a really good description of the relationship of horsepower and torque. The comment about the crossover point of horsepower and torque was really interesting. I never really thought about it but it is definitely true. Pretty cool.

I have to disagree with Steve's Schmidt's point that radians have a unit. A radian is a ratio. It just so happens that there is an angle that goes with it, but that is simply coincidental. When talking about radians you are referring to an arc LENGTH, not the angle related to that arc. So the radian measurement actually refers to the arc length relative to the radius of the arc. Because both of these have length units and the arc length is divided by the radius, all the units cancel and you end up with a unitless ratio.

Now, as Steve said, there really is no need to get into radians when breaking down horsepower. Since it's just a ratio there is no unit corruption. It just becomes a part of the bigger conversion factor.

Thanks,

Keith Menges

Here is a comment from Aivars Lelis,

Hi,

A simpler way to understand the relationship between torque and horsepower, and how to calculate it, may be as follows;

hp = torque (lb-ft) * rpm / 5252

Aivars Lelis

Here is a comment from Marty Rosalik,

For those hung up on radians per second we can stay in SI units and convert directly.

Torque(Nm) * Omega(rad/sec) = Watts.

Good luck finding a tachometer in radians per second.

Here is a comment from RJ67ChevySS,

I have to say that I belive that radians are in fact a unit. I belive that when you are refering to radians you are not refering to the length of an arc of a circle, but actually to the angle corresponding to that arc. A specific arc in a circle will change as the radius changes, yet the angle corresponding to that arc never changes. So only on a unit circle will 2 pi radians actually equal the length of the arc all the way around a circle, but on any circle 2 pi radians will be 360 degrees. Keith has a point in saying that radians don't have a unit because they are there own unit, just like inches and degrees. The best comparison I can make is that an inch is to a centimeter as a radian is to a degree.

Here is a comment from Jim Rennison,

The very easiest way to derive the HP formula is to remember that the distance around any circle measured in any units is 6.28 when we use a circle with a radius of one unit. In Watts classic definition, we have a HP as 33,000 lb*ft/minute. This is understood to be work done in a straight line. Since we already have a linear measurement contained in this definition, namely one foot, logically we can set the radius of our spinning shaft at one foot. This means we can then import torque figures in lb*ft right into the formula without any further calculations. Also, we need only the constant 2(pi) as a multiplier to converts watt's 'linear' HP into HP on a spinning shaft since we have smartly set the radius of that shaft at one foot. No need to worry about radians. So then HP=R(T)(2pi)/33,000 which can be written R(T)(2pi)/1 * 1/33,000 Dividing 33,000 by 2pi, 33,000/6.283185 =5252.1131. So then Hp = R(T)/5252. This is so simple and logical, you can derive the formula yourself if you forget.

Jimbo

Here is a comment from Tab Rasmussen,

I don't know if you have enough room on your site to post the attached article. Please feel free. I hope it helps makes sense of the ridiculous debate about horsepower and torque. This is really not as complicated and arcane as people think. The difference between the two is clear and simple, but folks cannot relate to the debate if it involves math and physics. Post my article and I believe more people will understand. Feel free, just cite me.

Tab Rasmussen Edwardsville, IL

Torque and Horsepower by Tab Rasmussen. This is a MS word document.

Thanks for your comments.