Energy in Simple Harmonic Motion

One total, two forms

For a mass on a spring the total mechanical energy is fixed by the amplitude alone and shuttles between potential and kinetic:

Two consequences are tested constantly. First, doubling the amplitude quadruples the energy but only doubles the maximum speed, since $v_{\max}=A\omega=A\sqrt{k/m}$. Second, the speed at any position follows directly:

A $0{,}5$ kg block on a $200$ N/m spring with $A=0{,}1$ m has $E=\tfrac12(200)(0{,}01)=1$ J, $\omega=\sqrt{200/0{,}5}=20$ rad/s and $v_{\max}=0{,}1\times20=2$ m/s.

Reading the energy graph

Plotted against position, potential energy is a parabola $\tfrac12kx^2$ and kinetic energy is the same parabola flipped, $E-\tfrac12kx^2$; their sum is a horizontal line at $E$. The two curves cross where $U=K=\tfrac12E$, which happens at $x=\pm\frac{A}{\sqrt2}\approx\pm0{,}71A$ — not at $x=\pm A/2$. At half the amplitude, only a quarter of the energy is potential and three quarters is kinetic. Plotted against time, both energies oscillate at twice the frequency of the motion, because each reaches its maximum twice per cycle.

The pendulum version

For a simple pendulum the potential energy is gravitational, $U=mgL(1-\cos\theta)$, measured from the lowest point. Released from rest at angle $\theta_0$, the bob's speed at the bottom is $v=\sqrt{2gL(1-\cos\theta_0)}$ — energy conservation, not $F=ma$, is the tool, because the tension does no work. The spring formula $\tfrac12kA^2$ does not apply; the small-angle approximation only says the period is amplitude-independent, not the energy.