A question in the derivation of work done

[ehabmozart]

In our formulation of work done by a Force we say it is ∫ F.dl ... My knowledge of integration means adding Infinitesimal parts. We write dl because it is the smallest part of the distance moved. My question is why have we written F and not dF... Thanks for clarifying!

 

[Doc Al]

In our formulation of work done by a Force we say it is ∫ F.dl ... My knowledge of integration means adding Infinitesimal parts. We write dl because it is the smallest part of the distance moved. My question is why have we written F and not dF... Thanks for clarifying!

The force doesn't get infinitesimally small. It remains finite as it acts over an infinitesimal distance.

 

[ehabmozart]

But not the full force is applied to this small part

 

[Doc Al]

But not the full force is applied to this small part

Sure it is.

Say I exert a force of 100 N over a distance of 1 meter. I'm exerting the full 100 N over every bit of that distance.

 

[ehabmozart]

What if the force is varying??

 

[jtbell]

What if the force is varying??

That's when you integrate. If the force is constant (and the object moves in a straight line), you simply have $W = \vec F \cdot \vec {\Delta l}$.

 

[Doc Al]

What if the force is varying??

As jtbell said, that's why you integrate.

But the force at any point is whatever it is, not some infinitesimal.

Say I push with some varying force for a distance of 1 meter. You can break that path up into as many small sections as you like. (When you integrate, you are making them infinitesimally small.) At any point in the path, the force has some value F(x) as it pushes through the tiny distance dx.

By analogy, think of calculating the distance traveled by something moving with some speed v:
X = V*t
Expressing it as a differential, you get dX = V*dt, not dX = dV*dt. Over the tiniest interval of time, the speed does not go to zero.