Confusing Units of Measure: Work, Power and Torque


Written by:
Joe Evans, Ph.D

Pumps & Systems, October 2007

Back in the early seventies, when I was in grad school, our government pledged to convert the U.S. measurement system to the metric system. A popular cartoon at the time showed a lab technician with a box of amputated human feet standing at the door of the supply room. The supply clerk was also holding a box, but his was full of volt meters. The caption was "Trading Feet for Meters." That was almost 37 years ago, and we still have most of those feet! I guess that I could say that we are still "inching" into the metric system. 

The SI system (Système international d'unités, or International System of Units) is the modern day version of the metric system, and the U.S. gets a lot of grief for not embracing its inherent transportability across international boundaries. It is the standard in our scientific community and its use is increasing in our engineering and industrial sectors, but it is still pretty unpopular with the proletariat. Why? Because the units of measure are totally unusable by most of us.

Take, for example, that simple quantity we know as work. Work, in a linear environment, is pretty straightforward (pun intended). Work (w) is defined as the product of the force (F) that is applied to an object and the distance (d) that object travels as a result of that force. So work is simply w = Fd.

In the good ole English system, force is measured in pounds and distance is measured in feet. If we lift a 100-lb box to a height of 10-ft, we perform 1000-ft-lb of work. Since work is directly proportional to both force and distance, a 10-lb box lifted to a height of 100-ft requires exactly the same amount of work.

Now, if we moved this box outside the U.S., SI would take over. First we would convert pounds to kilograms and feet to meters. Work in SI units is the joule, which is defined as a newton-meter. We all know a newton (named after Sir Isaac) is the amount of force required to accelerate a mass of 1-kg at a rate of 1-m/s2. Yeah, right!

It could be worse. Before SI, a unit of work could also be the erg, which is a dyne-centimeter. And a dyne is a gram-centimeter per second squared.

The English system equation for work tells us exactly how much work is performed and it does so in understandable units. In fact, gallons are easily converted to pounds, and we can use that same simple equation to evaluate the work done by a pumping system. What it does not tell us is how quickly that work gets done. When we lift that 100-lb box to a height of 10-ft, we perform 1000-ft-lb of work. It doesn't matter if it takes 10 seconds or 10 days, it is still 1000-ft-lb of work.

The rate at which work is done is referred to as power, and it is equal to the work performed divided by the time it takes to perform it. In simple terms, power = w/t. If it takes one minute to lift that 100-lb box to a height of 10-ft, the power required in English units is 1000-ft-lb/min. Again, pretty straightforward.

Now, you might think that the SI system, where power is measured in watts, would be equally understandable. Unfortunately, it isn't because the watt is defined as a joule per second. And that brings us back to the newton-meter.

Thanks to a fellow named James Watt, there is a more meaningful way to relate the watt to the ft-lb. In the late 18th century, he made some major improvements to the steam engine - improvements that made it a viable alternative to other sources of power. One of the power hungry applications at the time was coal mining, and most were powered by horses.

What Watt needed was a way to compare the power of his engine to that of a team of horses. The story goes that, through experimentation, he determined that the average horse could lift about 182-lbs to a height of 181-ft in one minute. (Several versions of this story are told, but the end result is always the same.)

Thus the power, or horsepower in this particular case, is 33,000-ft-lb/min. This quantity turns out to equal 745.7-joules/sec in the SI system. One joule per second was named a "watt" in his honor, so 1-hp is equal to approximately 746-W. In the U.S. we still rate an electric motor's power output in horsepower, while most other countries use kilowatts (kW).

Now, it is not difficult to visualize the work done in Watt's experiment. Just drop a long rope down a mine shaft, place it over a pulley, and hook the other end to a horse. The distance the horse walks times the weight he lifts is the work done. Measure how far he goes in one minute and you get horsepower.

Work and power can be a bit harder to visualize in a rotational environment. Take an electric motor, for example. How the heck do you measure force and distance?

In order to make some sense out of this we have to introduce a new term. If Watt's horse were to hold that weight in a static position, no work would be done, but the force holding it there would be similar to something we call torque. Torque can be considered the rotational counterpart of force and is defined as the product of force and the lever arm. Its SI unit is the newton-meter and its English equivalent is the foot-pound (ft-lb).

But wait, isn't that the unit for work? Yes it is, so torque is often called a pound-foot (lb-ft) or a foot-pound-force (ft-lbf) in order to differentiate it from work. Unfortunately, this definition is not always self-evident.

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