Oscillations

What you will understand

Predict what happens at the centre and the turning points, use the sign of velocity correctly, and calculate spring energy without confusing a fraction of energy with a fraction of speed. The model is ideal and undamped.

Quick answer: displacement does not specify direction

Measure $x$ from equilibrium. In the convention $x=A\cos\theta$, let $\theta=\omega t+\phi$, with $A\gt 0$ and $\omega\gt 0$. Then $v=-A\omega\sin\theta$ and $a=-\omega^2x$. A displacement can occur on both halves of a cycle; velocity chooses the branch. At $x=0$, speed is greatest and acceleration is zero. At $x=\pm A$, speed is zero and acceleration magnitude is greatest.

Same displacement, different velocity

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Use energy as an independent check

For an ideal horizontal mass–spring oscillator, $\omega=\sqrt{k/m}$. Choose spring potential energy zero at equilibrium: $E=\tfrac12 kA^2$, $U=\tfrac12 kx^2$ and $K=E-U$. Thus $U/E=(x/A)^2$ and $K/E=1-(x/A)^2$. Energy gives speed through $|v|=\omega\sqrt{A^2-x^2}$; it does not decide the sign of velocity.

Compare energy and speed fractions

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Common mistakes to catch

Zero velocity at a turning point does not imply zero acceleration. Convert centimetres to metres before calculating joules. Potential energy is zero at the chosen equilibrium reference, not universally at every “starting point”. At fixed $m$ and $k$, doubling amplitude doubles maximum speed and quadruples total energy; it leaves the ideal oscillator period unchanged.

Try it yourself 1: at a turning point

At $x=+A$ in ideal SHM, which statement holds? A. $v=0$ and $a=0$; B. $v=0$ and $a=-\omega^2A$; C. speed is maximum.

Try it yourself 2: change the amplitude

For the same ideal spring and mass, amplitude doubles. How do total energy and maximum speed change? A. Both double; B. energy quadruples and maximum speed doubles; C. both quadruple.

A useful check on your next visit

Cover the labels and redraw the centre and both turning points. Add velocity, acceleration, $K$ and $U$ at each. Recalculate the $x=0.6A$ energy and speed fractions. Use the distinction you missed to choose your next chapter-practice question.

Choose your next chapter task

Explain the mistake behind any self-check you missed, then use chapter practice to test the corrected idea.

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Sources and review

Technical review: Codex (automated). Sources checked on 23 October 2026. The diagrams, worked numerical models and self-checks are original teaching material. Human subject review is not claimed.