How To Calculate Mechanical Power

You can find​ mechanical power​ in use everywhere in the modern world. Did you ride in a car today? It used energy, either from fuel or a battery, to move an interconnected series of mechanical components – axles, gears, belts and so on – until, finally, that energy was used to spin the wheels and move the vehicle forward.

​Power​ in physics is a measure of the ​rate​ at which​ work​ is being performed over time. The word "mechanical" is merely descriptive; it tells you that the power is associated with a machine and the movement of different components like the drivetrain of a car or the cogs of a clock.

The mechanical power formula uses the same fundamental laws of physics that are used for other forms of power.

In order to make sense of the expression for mechanical power, it's helpful to lay out four interrelated terms: ​energy​, ​force​, ​work​ and ​power​.

•The ​**energy ​E​**​ an object contains is a measure of how much work it can do; in other words, how much motion it has the potential to create. It's measured in joules (J).
•A ​**force ​F​**​ is, in essence, a push or pull. Forces transfer energy between objects. Like velocity, force has both ​magnitude​ and ​direction​. It's measured in Newtons (N).
•If a force moves an object ​in the same direction​ it's acting, it performs​ work​. By definition, one unit of energy is needed to perform one unit of work. Because energy and work are defined in terms of one another, they are both measured in joules (J).
•​Power​ is a measure of the​ rate​ at which​ work is performed​ or ​energy is used  ​over time. The standard unit of power is the watt (W).

Because of the relationship between energy and work, there are two common ways of expressing power mathematically. The first is in terms of ​work​ ​W​ and ​time​ ​t​:

\(P=\frac{W}{t}\)

If you're dealing with linear motion you can assume that any force applied either moves an object forward or backward along a straight path in line with the force's action – think of trains on a track. Because the directional component basically takes care of itself, you can also express power in terms of a simple formula using ​force​, ​distance​, and ​velocity​.

In these situations, ​work​ ​W​ can be defined as ​force​ ​F​ × ​distance​ ​d​. Plug that in to the basic equation above and you get:

\(P=\frac{Fd}{t}\)

Notice anything familiar? With linear motion, ​distance​ divided by ​time​ is the definition for ​velocity​ (​v​), so we can also express power as:

\(P=F\frac{d}{t}=Fv\)

An Example Calculation: Carrying Laundry

OK, that was a lot of abstract math, but let's put it to work now to solve a sample problem:

​Your parents ask you to carry a 10 kilogram load of clean laundry upstairs. If it normally takes you 30 seconds to climb the stairs, and the stairs are 3 meters tall, estimate how much power will you need to expend to carry the clothes from the bottom of the staircase to the top.​

Based on the prompt, we known that time ​t​ will be 30 seconds, but we don't have a value for work ​W​. However, we can simplify the scenario for the sake of estimation. Rather than worry about moving the laundry up and forward on each individual step, let's assume you're simply raising it in a straight line from its starting height. Now we can use the ​P​ = ​F​ × ​d​ / ​t​ expression of mechanical power, but we still need to figure out the force involved.

In order to carry the laundry, you must counteract the force of gravity on it. Since the force of gravity is ​F = mg​ in the downward direction, you must apply this same force in the upward direction. Note that ​g​ is the acceleration due to gravity which on Earth is 9.8 m/s². With this in mind, we can create an expanded version of the standard power formula:

\(P=mg\frac{d}{t}\)

And we can plug in our values for mass, acceleration, distance and time:

\(P=(10\times 9.8)\frac{3}{30}=9.08\text{ watts}\)

So you'll need to expend about 9.08 watts to carry the laundry.

Our discussion has been limited to fairly straightforward scenarios and relatively simple math. In advanced physics, sophisticated forms of the mechanical power equation may require the use of calculus and longer, more complicated formulas that take into account multiple forces, curved movement and other complicating factors.

If you need more in depth information, the HyperPhysics database hosted by Georgia State University is an excellent resource.