If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Introduction to work review

Review the key concepts and skills for energy and work.  Understand how work is the area under a force vs. displacement graph and how force and displacement produce work.

Key terms

Term (symbol)Meaning
Energy (E)Measurement of the ability to do work. SI unit is joule (J).
Work (W)Change in energy by the transfer of energy from one system to another. Scalar quantity with units of joules (J).
Joules (J)SI unit for energy. Applying a net force of 1N to an object over a displacement of 1m requires 1J of energy. 1J=1N1m=1kgm2s2


EquationSymbol breakdownMeaning in words
W=FdcosθW is work done on an object, F is the magnitude of force on the object, d is the magnitude of the object’s displacement, and θ is the angle between the vectors of force F and displacement d.Work is the product of an object’s displacement and the component of force exerted parallel to the direction of the object’s motion.
W=ΔEW is work and ΔE is the change of energy.Work is the change of energy for a system.

How to find work from a Force vs. displacement graph

The force applied to an object can be graphed as a function of the position of the object. Work is the area under the curve of the force vs. position graph. Areas above the position axis are positive work and areas below the axis are negative work. If the force is not constant, we can divide the graph into sections with simpler shapes and add up the work in each section. To find the total work done on the object in the force vs. position graph in Figure 1 over displacement d1+d2, the areas of A1 and A2 can be added together.
Graph of Force vs. displacement. F is vertical axis and x is horizontal axis. At a height of F_o a blue line extends on the graph horizontally for a distance of d_1. This forms a red-shaded area labeled A_1 that is a rectangle. From the end of the horizontal line, another blue line extend diagonally down towards a mark d_2 further long the x-axis. This forms a purple-shaded triangle labeled A_2.
Figure 1. Work is the area under a force vs. displacement graph. This graph can be analyzed as two separate areas.
A1 is a rectangle of height Fo and width d1. A2 is a triangle of height Fo and base d2. The total work done on the object over d1+d2 is
For worked examples of finding work from a force vs. displacement graph, watch our video about calculating work from force vs. displacement graphs.

Common mistakes and misconceptions

1) People forget that forces acting perpendicular to displacement do zero work. Since cos90=0, no work occurs when force F and displacement d are perpendicular even though a perpendicular force can change the object’s direction of motion. For example, if a box is pulled across the level floor, the floor does zero work on the box even though the box is moving. This is because the normal force is perpendicular to the horizontal displacement of the box.
2) People forget what the sign of work means. Positive work on a system means it receives energy from its surroundings. Negative work on a system means it gave energy to its surroundings. Negative work occurs when the force has a component in the direction opposite the displacement.

Learn more

For deeper explanations of work, see our video with example problems about work.
To check your understanding and work toward mastering these concepts, check out the exercises in this tutorial: work from force vs. position graphs and calculating work from a force.

Want to join the conversation?