The change in gravitational
potential energy is defined as the *work done against gravity*
in moving an object from initial to final position. The work
depends *only* on the initial and final positions, not the
path taken.

*W _{g}* = − ∫

Note that since *y = 0* can
be set anywhere, the zero of gravitational potential energy is *completely
arbitrary.* Therefore it cannot be used unless you specify
at what spot it is taken to be zero!!

The sum of kinetic energy and potential energy remains constant if no other forces than gravity do work.

If only one conservative force is
doing work, and no non-conservative forces act, then because of
the way we have defined KE and PE, whatever dependence the PE *U*
has on position, the dependence of the KE *K* is
guaranteed to mirror it in the opposite sense. Thus *E = K +
U* must be a constant independent of position or any other
property of the system.

Since *E = K + PE,* a point
where *E = PE* is called a classical turning point. And
on a plot, for example
of *PE = *(1/2)*kx ^{2},* it is easy to see
graphically what