# Why is the work done by moving an object up vertically not greater than mgh

Watching Walter Lewin's classical mechanics. In lecture 11 he says when moving object up vertically distance h, the work done by gravity is -mgh, which makes sense. But then he said the work done by him is obviously mgh. Why is that the case? Doesn't the force he exerts need to be greater than mg in order to overcome the force of gravity? I.e. I would expect the work done by him to be (mg+ma)h. Thank you for your help.

Doesn't the force he exerts need to be greater than $$mg$$ in order to overcome the force of gravity?

To an extent, yes. But you can assume that in the limit, it can be arbitrarily close to $$mg$$.

Further, he supplies the constraint that the object reaches the point after a height $$h$$ with zero velocity. Therefore if you assume that at some point you exerted a force greater than $$mg$$ to accelerate the object, you must also have allowed the force to be less than $$mg$$ to decelerate the object.

I would expect the work done by him to be $$(mg+ma)h$$

KE is unchanged (the velocity at the start and the end is identical), and the work done by gravity is $$-mgh$$. Therefore there must have been something doing exactly $$mgh$$ work on the object, and that is from the person doing the lifting. Any work done to accelerate at the start the lift is repaid when the object is stopped.

• Yes you are absolutely correct. I thought that to be the case at first but later on around 13:30 he has another example of taking any path from point A to point B, and I realize now that the object is not moving at A and B, whereas I had assumed A and B could be any two points along a path, and I think that's what makes all the difference. Thanks for your answer. Commented Jan 20 at 1:42

$$W_{person} = F_{applied} \times h$$

$$h$$ is the distance the object moves throughout the application of the force.

The crucial point here is that the force applied by the person only needs to be equal to the force of gravity to lift the object at a constant velocity. If the force applied is equal to or greater than mg, the object will accelerate upwards with increasing velocity. Since at all times the object is in contact with the person, it is a fair assumption that no upward velocity is imparted in the object.