Work in Physics - What Is It, and How Do We Calculate It?

YouTube Screenshot (https://www.youtube.com/watch?v=7r4ZS-LYzGs)

By Andrew Bennett

Work is a tricky term to define because it reaches into considerations of both forces and energy. It is also a quantity that might be easier to understand through equations than word-based definitions.

Definition of Work in Words

Often, we say that work is the transfer of energy from one system to another by applying a force over some distance. Then, of course, we have to understand energy to understand work. Since we often define energy as the ability to do work, we end up with a somewhat circular definition. What we should recognize right away, though, is that work involves both forces and energy. So, it can sometimes be used as a bridge between two ways of thinking about a scenario.

Equations for Work Involving Forces

On the forces side of things, we define work as the dot product of the displacement vector (r) and the force vector (F), or:



A dot product is one way to multiply two vectors that produces a scalar (non-vector) answer. The answer can be calculated as

where theta is the angle between the displacement and force vectors. From this form of the equation, we can see that work can be positive when the angle is between 0 and 90 degrees, negative when the angle is between 90 and 180 degrees, and zero when the angle is exactly 90 degrees. 

In general, we say that work is positive when the force is at least somewhat in the same direction as the object moves. However, work is negative when the force is at least somewhat in the opposite direction as the object moves. Then, work is zero when the force is perpendicular to the direction the objects moves. Also, work will be zero when the displacement is zero, meaning that no work is done if the object doesn't move or ends back where it started.

It is worth noting that each force acting on an object can do work on the object, so you might need to calculate the work done by each separate force acting on the object. You could also calculate the net work done on the object by adding up the work done by each force or by determining the net force acting on the object and calculating work done by the net force. (Both methods give the same answer.)

Physics Work Problems Video

Be sure to watch this video for an overview of these types of calculations:

Work from a Graph of Force vs. Position

The equations above are actually special cases of a more general equation for work that involves calculus. The versions we see here can only be used with force stays constant.

What do we do when the force changes? The short answer is: calculus. Don't worry, though: If you can calculate the area of triangles and rectangles, you can do this type of calculus.

Physics and Calculus Work Problems Video

Check out this video for more information:

Equations for Work Involving Energy

Work can also be defined as a transfer of energy from one system to another. In fact, the amount of work done on a system will be exactly equal to the change in energy for the system. If energy isn't added or taken in any other form, we can use the equation

to solve problems. Some problems that we see (especially early on in physics) are simplified further such that only one type of energy is changing. For example, if a motor was doing work on a car to accelerate it on a flat, horizontal road, the only type of energy the car would gain is kinetic energy. So, we could rewrite the above equation as:

.

Work and Kinetic Energy Videos

This type of calculation is explained and demonstrated in this video:

Once we get information about kinetic energy, it is a fairly simple step to use the kinetic energy equation to find out something about an objects speed or mass.  We could also use known changes to speed to determine the amount of work done, as shown here:

With the addition of these types of equations, we can link information involving forces to information about energy and solve problems that straddle these two topics, as demonstrated in this video:

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