8.3.1 Derivation of TE Formula

To produce an oscillation with amplitude x₀, the system must first be displaced from its equilibrium position to $\displaystyle x={{x}_{0}}$ . This requires an external force to pull or push against the restoring force. The work done by this external force corresponds to the (total) energy of oscillation, TE.

Recall that $\displaystyle a=-{{\omega }^{2}}x$ and the restoring force is $\displaystyle \displaystyle {{F}_{{net}}}=-m{{\omega }^{2}}x$ . Since the external force must match the restoring force at every position, $\displaystyle \displaystyle {{F}_{{ext}}}=m{{\omega }^{2}}x$ at every position.

Since the area under the F–x graph is work done, we can derive the formula for TE to be

$\displaystyle TE=\frac{1}{2}({{x}_{0}})(m{{\omega }^{2}}{{x}_{0}})=\frac{1}{2}m{{\omega }^{2}}{{x}_{0}}^{2}$

If the restoring force is the only force acting on the system (i.e. no other dissipative forces), then the system will oscillate forever (after it has been set in motion). During the oscillation, the restoring force causes energy to be converted between the kinetic energy KE and potential energy PE of the system. From the principle of conservation of energy, we know that,

$\displaystyle TE=KE+PE$