In my lecture today my professor briefly mentioned that force is the derivative of energy...

logiski9s

logiski9s

Answered

2022-07-16

In my lecture today my professor briefly mentioned that force is the derivative of energy but I did not really get what he meant by that. I tried to express it mathematically:
d d t K E = d d t ( 1 2 m v 2 ) = m v d v d t
This looks really close to Newton's second law F = m a but there is an extra "v" in there. Am I missing something here?

Answer & Explanation

tilsjaskak6

tilsjaskak6

Expert

2022-07-17Added 14 answers

For conservative systems, it is true that the force can be expressed as minus the gradient of the potential energy:
(1) F ( x ) = V ( x ) ,
which can be though of as the defining property of a conservative system.
The gradient reduces for one-dimensional systems to the derivative with respect to the space coordinate, i.e. you have in this simple case
(2) F = d V d x .
Taking as an example the case of a mass m in the gravitational field of the earth, you have the potential energy
(3) V ( z ) = m g z ,
where z is the distance from the ground. The force in the z direction is then given by
F g = d V ( z ) d z = m g ,
which is what you would expect.

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