This is Liao Hui, not doing any work. He did work to get the weight over his head, but despite the tremendous force he’s applying to this 348 kilogram [*Update: Thanks to commenter Ducklike for correcting this to 158 kg*] weight he’s not doing any work in this picture. The weight is stationary.

Work in physics is a term of art meaning *force through a distance*. The reason for that definition of work is that this definition coincides perfectly with the energy transferred in the process. Work results in a change in energy. When Liao Hui did work to move the weight to its highest position, chemical energy from his body was transferred into his muscles which did work to the weight. As the work against gravity caused the weight to increase in height, it gained gravitational potential energy. And finally when Liao Hui let go of the weight, it fell back to earth: gravity did work in the opposite direction which increased the kinetic energy of the weight until it hit the ground, releasing the energy as heat and the sound of the impact.

Force through a distance. Mathematically that means work in joules is the force in newtons multiplied by the distance in meters.

I’d figure out how much work was done by the weightlifter but I don’t know how far it was lifted. Probably between 1 and 2 meters, so a few thousand joules.

As always in physics, we can take the basic concept and make it more general. What if the force isn’t pushing in the same direction as the motion? For instance, say your friends are pushing a car that has run out of gas. You decide to be a joker and start pushing the car from the side, at right angles to the direction of motion. That won’t change the car’s speed and you aren’t adding or subtracting any energy to the car, so you can’t be doing any work. It’s because you aren’t pushing in the direction of motion. Call the angle between the force and the motion by the Greek letter theta:

The work is proportional to the cosine of the angle between the force and the displacement. If the angle is zero, we just get the Fd relationship we had initially. We can save some space by writing the relationship in vector notation, which is what the last part of that equation above means.

But even this isn’t quite enough to sum up the totality of possibility. The force might not be constant, and the displacement might not be along a straight line. You could have a force with constantly changing direction and magnitude pushing an object along some weird curvy path. So the final generalization is to say the total work is the sum of each little bit of work done along the path by whatever the force and its angle happen to be at that but of path. This is written using an integral.

Here we call the path by the label C, and we call the distance by the variable s because d is already taken by the notation of the integral itself. That allows us to find the general work done over a general path.

It can be tricky to calculate that integral. But in many cases, nature works in such a way as to help us out. If the work is being done against a conservative force, the integral is just the change in potential energy. A weightlifter can use whatever weird method he wants to get the weight up, but that integral will just turn out to be the change in potential energy mg(Δh).

This fact that mechanics can be thought of in terms of force or energy equally well turns out to be one of the most important concepts in physics. And while it might not help anyone actually pick that weight up, it’s a good mental exercise to go along with the physical exercise.