alternative method to obtain potential energy
المؤلف:
Professor John W. Norbury
المصدر:
ELEMENTARY MECHANICS & THERMODYNAMICS
الجزء والصفحة:
p 103
15-12-2016
2417
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!
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