*Joe Evans is responsible for customer and employee education at PumpTech, Inc., a pumps & packaged systems manufacturer & distributor with branches throughout the Pacific Northwest. *

### As a follow up on my AC Motors series, I thought it would be a good idea to provide a short overview of work, power and torque as it applies to the AC motor. I will provide a reference to a more in depth study at the end of this article.

## Work

The AC motor is a machine designed to transform electrical energy into mechanical energy so that work can be performed.

Work (w), in a translational (linear) system, is defined as the force (f) applied to some object multiplied by the distance (d) that it travels. In the English system force is measured in pounds and distance is measured in feet.

The equation for work, which is quite straight forward (no pun intended), is:

w = fd

If you lift 100 pounds to a height of 10 feet, you perform 1,000 lb-ft of work.

The same amount of work is performed if you lift a 200 pound object to a height of 5 feet or, for that matter, a 50 pound object 20 feet.

Work is a somewhat unfortunate term because for work to be performed, we must actually move an object in a direction that is opposite the force acting upon it.

For example, if you lift a suitcase off the floor you have performed work because the force you applied overcame the force of gravity that was holding it to the floor.

Carrying it across the room, however, is not work because you do not move in a direction that opposes the force acting upon it. However, try telling that to someone with a thirty pound suitcase in each hand. He or she will expend energy but will perform no work.

The equation for work tells us how much work is done, but it says nothing about how quickly it gets done. If you carry a 50 pound object up a flight of stairs that are 10 feet high, you will perform 500 lb-ft of work. It makes no difference if you do it in five seconds or five days, the same amount of work is performed.

## Power

The rate at which work is done is power. Power (p) is equal to work (w) divided by the time (t) it takes to perform it. The equation, also very straight forward, is:

p = w / t

In the late eighteenth century, James Watt 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 in Scotland at the time was pumping water from coal mines.

The pumps were powered by horses, and Watt needed a way to relate the power of his engine to that of a team of horses.

Through experimentation, he reportedly determined that the average horse could lift 150 pounds to a height of 220 feet in one minute. The work performed then, is 33,000 lb-ft (w = fd). Power, or in this case horsepower, is 33,000 lb-ft/min. This rather cumbersome number is equal to 745.7 joules/sec in the MKS system.

One joule/sec was called a watt in his honor. Therefore, one horsepower is equal to approximately 746 watts. In the United States we rate a motor's power in horsepower. In most other countries, it is the kilowatt (KW).

## Torque

Torque is defined as the force that gives rise to rotational motion. It is also the result of rotational motion. Torque (t) is equal to force (f) times the radius (r) through which it acts and the angle (Ø) at which the force is applied. The equation used to calculate torque is:

T = f (r sin Ø)

Torque in a rotational system is analogous to force in a translational system. The straight line distance of the translational system, however, is replaced with an angular quantity.

For any given horsepower, torque varies inversely with rotational speed.

For example, a 100 horsepower motor operating at 3,600 rpm produces a torque of approximately 150 lb-ft. At 1,800 rpm, torque would be about 300 lb-ft and at 1,200 rpm about 450 lb-ft. This is exactly what one would expect since horsepower (power) is the rate at which work is done.

If a 1,800-rpm motor is to accomplish the same amount of work in the same amount of time as one rotating at 3,600 rpm, it must do twice the work per rotation.

It is for this reason that the shaft diameter of a 900-rpm motor is larger than that of the 3,600 rpm model even though they produce the same horsepower.

Lower rpm motors use larger diameter shafts to accommodate the higher torque required to do the same amount of work in fewer rotations.

Earlier, horsepower was defined as w / t and expressed in lb-ft or watts. It can also be defined in terms of torque (t) and speed (rpm).

HP = (t x rpm) / 5,250

Torque can also be expressed in terms other than force and the radius through which it travels.

Rearranging the previous equation we get:

t = (HP x 5,250) / rpm

These two equations are probably more useful in daily work than the previous ones.

For a more detailed review of work, torque and power see my "Confusing Units of Measure," *Pumps & Systems* October 2007.

*Pumps & Systems,* August 2011