A Level Physics: Oscillations — Revision Notes

Condensed for the final weeks. For the full explanation, use the Oscillations study guide, and test yourself with the practice questions.

Defining SHM

Motion in which the acceleration is proportional to the displacement from a fixed point and always directed towards that point — the defining “restoring” behaviour that makes SHM oscillate rather than move off indefinitely.

a = -omega^2 x

The minus sign is the definition — it encodes “directed towards equilibrium”. Omitting it loses the mark even if the rest is right.

Equations

angular frequency   omega = 2 pi f = 2 pi / T

displacement        x = x0 sin(omega t)     (starting at equilibrium)
                    x = x0 cos(omega t)     (starting at maximum)

velocity            v = +/- omega sqrt(x0^2 - x^2)
maximum velocity    v_max = omega x0        (at x = 0)
maximum accel.      a_max = omega^2 x0      (at x = +/- x0)

Worked example. Amplitude 0.050 m, angular frequency 4.0 rad/s.

v_max = omega x0 = 4.0 x 0.050 = 0.20 m/s

Where each quantity peaks

At equilibrium (x = 0)	At maximum displacement (x = ±x₀)
Velocity maximum	Velocity zero
Acceleration zero	Acceleration maximum
Kinetic energy maximum	Potential energy maximum
Potential energy zero	Kinetic energy zero

Velocity and acceleration are 90° out of phase; acceleration and displacement are 180° out of phase.

Energy in SHM

total energy   E = 1/2 m omega^2 x0^2       (constant)
kinetic        Ek = 1/2 m omega^2 (x0^2 - x^2)
potential      Ep = 1/2 m omega^2 x^2

Both Ek and Ep vary as x², so their graphs against displacement are parabolas — and their sum is a horizontal line, since total energy is conserved throughout the oscillation.

Free, damped and forced oscillations

A free oscillation occurs at the system’s own natural frequency, with no external driving force and, ideally, no energy loss. A damped oscillation loses energy (usually to resistive forces such as friction or air resistance), so its amplitude falls over time. A forced oscillation is driven by an external periodic force at a chosen driving frequency, which need not match the natural frequency.

Damping

Type	Behaviour
Light	Amplitude decays exponentially over many oscillations
Critical	Returns to equilibrium in the shortest time without oscillating
Heavy (over)	Returns slowly, no oscillation

Critical damping is the one used in car suspension and analogue meters — a car suspension damped too lightly would keep bouncing after every bump, while one damped too heavily would respond sluggishly.

Damping reduces amplitude but leaves the period essentially unchanged for light damping — a common point examiners test by asking what changes and what stays (approximately) the same.

Resonance

When the driving frequency equals the system’s natural frequency, energy transfer is maximum and amplitude peaks.

Increasing damping lowers the peak amplitude and shifts it to a slightly lower frequency, while broadening the curve — a heavily damped system barely shows a resonance peak at all.

Examples: tuned circuits (selecting a radio station), MRI (nuclei driven at their resonant precession frequency), musical instruments (a sound box resonating at the note’s natural frequency) — and the destructive cases such as bridges and buildings shaken at their natural frequency by wind or an earthquake.

Exam traps

• Omitting the minus sign in a = −ω²x.
• Confusing where velocity and acceleration are maximum.
• Using degrees when ω is in rad/s.
• Saying damping changes the period noticeably — for light damping it does not.
• Describing critical damping as “the fastest return” without “without oscillating”.
• Forgetting total energy in SHM is constant.
• Calling every oscillation a “free” oscillation — a free oscillation specifically has no external driving force, unlike a forced oscillation.
• Confusing the driving frequency with the natural frequency — resonance is defined by the two being equal, not just “high”.
• Reading sin ωt and cos ωt as interchangeable — which one applies depends on whether the object starts at equilibrium (sin) or at maximum displacement (cos).

Self-test

1. Define simple harmonic motion.
2. Where in the cycle is acceleration greatest, and where is velocity greatest?
3. A pendulum has amplitude 0.05 m and ω = 4 rad/s. Find v_max.
4. What distinguishes critical from heavy damping?
5. What happens to the resonance peak as damping increases?
6. Distinguish a free oscillation from a forced oscillation.
7. A mass–spring system oscillates with amplitude 0.030 m and angular frequency 6.0 rad/s. Find its maximum acceleration.

Answers: 1. Motion in which acceleration is proportional to displacement from a fixed point and always directed towards that point (a = −ω²x). 2. Acceleration is greatest at maximum displacement; velocity is greatest at the equilibrium position. 3. v_max = ωx₀ = 4 × 0.05 = 0.2 m/s. 4. Critical damping returns the system to equilibrium in the shortest possible time without oscillating; heavy damping returns it more slowly, also without oscillating. 5. The peak amplitude falls, the peak shifts to a slightly lower frequency, and the curve broadens. 6. A free oscillation occurs at the system’s natural frequency with no external driving force; a forced oscillation is driven by an external periodic force at a chosen frequency, which may differ from the natural frequency. 7. a_max = ω²x₀ = 6.0² × 0.030 = 36 × 0.030 = 1.08 m/s².