Relationship between potential energy and force

[Aaron Wong]

Hi,
I wonder which steps of my following deduce are wrong
dU=-dW
dU=mg dh
dW=Fdx
thus, mgdh=-Fdx (dx=dh)
then, F=-mg which is a constant in most of situation
However, F does change in some situation.

I am confused about this.

 

[Simon Bridge]

Sticking to 1D, if you have potential energy distribution U(x) so U is explicitly dependent of the position x ... then:
F = -dU/dx ... so you see F will depend on x if dU/dx depends on x.

Constraining motion to the x axis, the work done by force F is given by:
W = Fx so dW/dx = x.dF/dx + F (by the product rule) so dW = x.dF + F.dx and substituting above gives:
dW = x.dF - dU which gives a 2nd order DE in U.
Thus dW = -dU everywhere only in the situation where d²U/dx² = 0 ...
ie. The expression you started with is a 1st order approximation only (expand U(x) as a power series and see.)

Notice that F=mg means that dU/dx = -mg and d²U/dx² = 0 so the relation is exact, and you will see that dU/dx does not depend on x (see first sentence above).

Try for a harmonic oscillator ... $U(x)=\frac{1}{2}kx^2 \implies F=?$ ... here I have chosen U(x):U(0)=0 for you.