why does work depend only on displacement?

Question

i don't understand why work depends only on displacement...work is scalar so shouldn't it depend on exactly how much distance is traveled, not just the distance and end point? is there some sort of mathematical way to prove why it only depends on displacement? please explain!

Answer 1

Work is equal to Force x Displacement so displacement and force are the only factors involved in work. If you think about the work-kinetic energy theorem which states that work is equal to a change in kinetic energy, the only way to change your kinetic energy is through motion which Newton says requires a force, which in turn gives it a displacement.

Answer 2

Colin's answer is completely devoid of meaning. The other isn't better.

Work does not depend only on displacement, except if you have a conservative field of forces (like gravity) in which case only the level difference matters. But that is a special case.

In other cases, the work done in a very small displacement is the scalar product of the force and the displacement vector. dW = F.dl = F*dl*cos(theta) with theta the angle of the force and vector dl. And to have total work, you must integrate (sum) that along the whole path. So if the force is constant (this is still a conservative field), it will give you something like the displacement in the direction of the force (times force). If the force is constant in modulus but always tangent to the path, you'll have the full path length (times force) as you suspected. Your question is very good.

Answer 3

BECAUSE it is scalar. If you move an object 30 m, and then back 10 m, the work done will be that required to move it 20 m, because when you move it backwards 10 m you are doing negative work. As far as your muscles are concerned of course, you've done 40, but the object doesn't care about that :p.