Prove f=-dv/dx for conservative force.
Answers
Answer:
There are some good answers already. I will try to offer a simpler but less rigorous /less general proof.
You make some change which involves doing work on an object. (It could be pushing a ball up a slope)
Work Done = Force on object (F) x distance moved in direction of object (-delta x).
=mg(-delta x) …….(1)
Change in GPE = delta U x mass (m) ……(2)
Conservation of energy:
work done on work done + change in GPE =0
Work done = 0- change in GPE …. (subst in eq (1) and (2))
mg(delta x) =0- delta U x m …. (cancel m)
g delta(x)= - delta U
g= -delta U/ delta (x) … take limits g= -dU/dx where g is the gravitational field strength.
Note: that if instead of F=mg you used F=qE for electric charges , it would be the same but you would cancel q rather than m.
I have just noticed that the question is wrong. It is the field srength which is equal to minus the potentail gradient, not the force.