alternative method to obtain potential energy

Potential energy is defined through

Let's just ignore the vectors for the moment and write

Thus we must have

To see this write

(cf. Fundamental Theorem of Calculus).

For gravity we have  or F = -mg and for a spring we have  or F = -kx. Thus instead of working out the integral to get U, just ask what U will give F according to .

Example For gravity F = -mg, derive U without doing an integral.

Solution For gravity dr ≡ dy. The question is what U will give

The answer is U = mgy. Let's check:

which is the F we started with !

Example For a spring F = -kx, derive U without doing an integral.

Solution For a spring d ≡ dx. The question is what U will give

The answer is . Let's check

which is the F we started with!