A block on a spring is the model oscillator. Exam questions chain the period from Period of a Mass on a Spring with the energy ideas of Energy in Simple Harmonic Motion, then ask about timing. Each problem here is one short scene and a chain of questions. Answer them in order: each later question uses what you found before, so a slip early on carries through. Keep full precision between steps and round only at the end.
Key idea
$$T = 2\pi\sqrt{\frac{m}{k}}, \qquad \omega = \sqrt{\frac{k}{m}}, \qquad v_{\max} = A\omega, \qquad a_{\max} = \frac{kA}{m}.$$ Released from $x = A$: $x = A\cos(\omega t)$. The rest point comes after $T/4$; halfway ($x = A/2$) after $T/6$, because the block starts slowly.
Worked example: Pull and let go
A $0.5\ \text{kg}$ block on a $50\ \text{N/m}$ spring is pulled $8\ \text{cm}$ and released. Find (1) the period and (2) the greatest speed.
- (1) $T = 2\pi\sqrt{0.5/50} = 0.628\ \text{s}$.
- (2) $\omega = \sqrt{50/0.5} = 10.0\ \text{rad/s}$, so $v_{\max} = (0.08)(10.0) = 0.800\ \text{m/s}$.
Answer: $0.628\ \text{s}$, $0.800\ \text{m/s}$
Common mistake
Thinking a bigger pull gives a longer period. $T = 2\pi\sqrt{m/k}$ has no amplitude in it: a bigger swing is covered faster. Check with the formula before trusting your instinct.