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Energy conservation is one of the most powerful tools for SHM problems. Total energy $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2A^2$ is constant throughout the motion. At displacement $x$: kinetic energy $KE = \frac{1}{2}k(A^2-x^2)$ and potential energy $PE = \frac{1}{2}kx^2$.

Key energy milestones: at $x = A/\sqrt{2}$, $KE = PE = E/2$. At $x = A/2$, $KE = 3E/4$ and $PE = E/4$. In general, at displacement where $KE = nPE$: $x = A/\sqrt{n+1}$. Both KE and PE oscillate sinusoidally at frequency $2\omega$(twice the SHM frequency), always remaining non-negative. Their time averages are each$E/2$. Energy is proportional to$A^2$: doubling amplitude quadruples energy. Critically, energy does NOT depend on mass for a given spring and amplitude ($E = \frac{1}{2}kA^2$), though it does depend on mass when expressed as$E = \frac{1}{2}m\omega^2A^2$. The energy method is often the cleanest approach for complex systems: write the total energy as$E = \frac{1}{2}m_{\text{eff}}\dot{x}^2 + \frac{1}{2}k_{\text{eff}}x^2$, then read off$T = 2\pi\sqrt{m_{\text{eff}}/k_{\text{eff}}}$.$