Velocity, Acceleration, and Extrema
Differentiating sinusoidal displacement gives v = A omega cos(theta) and a = -omega^2 x, so v^2 = omega^2(A^2 - x^2) for ideal SHM.
Why this shows up in the exam
Finding amplitude from instantaneous data · Interpreting velocity-displacement plots · Comparing maximum speed and acceleration
Learn the idea
Speed is greatest at equilibrium, while acceleration magnitude is greatest at the turning points. The oscillator trades restoring pull for speed: at the centre the pull vanishes but motion is fastest, and at an extreme it stops while the pull is strongest.
🧠 Memory hook: Centre: fastest. Ends: strongest acceleration.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- v² = omega²(A² - x²) — speed at displacement x
- v_max = A omega — maximum speed at equilibrium
- a_max = A omega² — maximum acceleration magnitude at an extreme
How to approach it
- 1Identify the requested instantaneous or maximum quantity
- 2Use the displacement identity when time is absent
- 3Check values at x = 0 and x = plus or minus A
Common slip-ups that cost marks
- •Attaching the wrong sign to speed
- •Putting maximum speed at maximum displacement
- •Mixing centimetres and metres in ratios with units
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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