SHM Total Energy
Gives the constant total mechanical energy of an undamped oscillator: E = ½kA²
The equation
What it solves
Gives the constant total mechanical energy of an undamped oscillator: E = ½kA². Energy oscillates between purely kinetic (at equilibrium) and purely potential (at amplitude), but the total stays fixed.
When to use it
Finding maximum speed (v_max = A·ω = A√(k/m)), finding amplitude from total energy, or computing the speed at an arbitrary position via KE = E − ½kx².
When NOT to use it
E = ½kA² applies only to undamped oscillators. In a damped system, energy decreases exponentially; in a driven system, the source continually replenishes energy.
Common mistakes
Using ½mv² for total energy without including the potential energy ½kx² — the sum E is constant only when both are included. Confusing A (amplitude) with x (instantaneous displacement). Forgetting to square the amplitude: E scales as A², not A.