An oscillator trades energy back and forth between motion and storage. Conservation of mechanical energy turns that trade into quick answers about speeds, without any graphs or timing, for any simple harmonic oscillator.
Why it works
A spring stretched by $x$ stores $U = \tfrac{1}{2}kx^2$. With no friction, the total $\tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2$ never changes. At a turning point the mass stops for an instant, so all the energy is stored: $E = \tfrac{1}{2}kA^2$. At equilibrium the spring stores nothing, so all of it is kinetic: $\tfrac{1}{2}mv_{\max}^2 = \tfrac{1}{2}kA^2$. In between, the energy is shared.
Key idea
$$\tfrac{1}{2}mv^2 + \tfrac{1}{2}kx^2 = \tfrac{1}{2}kA^2.$$ The speed is greatest at equilibrium, $v_{\max} = A\sqrt{\dfrac{k}{m}}$, and zero at the turning points. At any position, $v = \sqrt{\dfrac{k}{m}\left(A^2 - x^2\right)}$.
Worked examples
Worked example: Total energy and top speed
A $0.50\ \text{kg}$ block on a spring ($k = 50\ \text{N/m}$) oscillates with amplitude $0.10\ \text{m}$. Find its total energy and greatest speed.
- $E = \tfrac{1}{2}kA^2 = \tfrac{1}{2}(50)(0.10)^2 = 0.25\ \text{J}$.
- $$v_{\max} = A\sqrt{\dfrac{k}{m}} = (0.10)\sqrt{\dfrac{50}{0.50}} = 1.0\ \text{m/s},$$ at equilibrium.
Answer: $0.25\ \text{J}$; $1.0\ \text{m/s}$
Worked example: Speed partway
For the same block, how fast is it moving at $x = 0.06\ \text{m}$?
- Stored: $\tfrac{1}{2}(50)(0.06)^2 = 0.09\ \text{J}$, so $K = 0.25 - 0.09 = 0.16\ \text{J}$.
- $v = \sqrt{\dfrac{2(0.16)}{0.50}} = 0.80\ \text{m/s}$.
Answer: $0.80\ \text{m/s}$
Worked example: Where the energy splits evenly
At what position is $K = U$?
- Then $U$ is half the total: $\tfrac{1}{2}kx^2 = \tfrac{1}{2}\cdot\tfrac{1}{2}kA^2$, so $x = \dfrac{A}{\sqrt{2}} \approx 0.71A$.
- For $A = 0.10\ \text{m}$ that is $0.071\ \text{m}$, not halfway, because $U$ grows as $x^2$.
Answer: $x = A/\sqrt{2}$
Common mistake
Thinking the speed is greatest at the turning points. The turning points are where the mass stops and reverses, so the speed there is zero; what is greatest there is the stored energy (and the acceleration). The speed peaks at equilibrium, where all the energy is kinetic. Check with energy: at $x = \pm A$ the spring already holds the whole $\tfrac{1}{2}kA^2$, so nothing is left for motion.