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h4. Elastic Potential Energy

Assuming an object attached to a spring that obeys Hooke's Law with the motion confined to the _x_ direction, it is customary to choose the coordinates such that _x_ = 0 when the object is in a position such that the spring is at its natural length.  The force on the object from the spring is then:

{latex}\begin{large}\[ \vec{F} = - kx \hat{x} \]\end{large}{latex}

It is also customary to make the assignment:

{latex}\begin{large}\[ U(0) \equiv 0\]\end{large}{latex}

Thus, the potential can be defined:

{latex}\begin{large}\[ U(x) = U(0) - \int_{0}^{x} (-kx)\:dx = \frac{1}{2}kx^{2}\]\end{large}{latex}

For an object moving under the influence of a spring only, the associated potential energy curve would then be:

POTENTIAL ENERGY CURVE

ForThe suchgraph aindicates motion,the then, there ispresence of one stable equilibrium point at _x_ = 0.