Edexcel IAL Physics: Oscillations — Revision Notes

Condensed for the final weeks. For the full explanation, use the Oscillations study guide.

Simple harmonic motion

Definition — both parts are required:

Acceleration is proportional to displacement from the equilibrium position and always directed towards it.

a = -omega^2 x
x = A cos(omega t)   or   A sin(omega t)
v_max = omega A        a_max = omega^2 A
omega = 2 pi f = 2 pi / T

Background — beyond the specification, not among outcomes 143-145 or the supplied formulae: the velocity-displacement relation, v = ±ω√(A² − x²), gives the speed at any displacement x directly, without needing time t.

The minus sign is the definition. It encodes “towards equilibrium”, and dropping it means you have not stated SHM. The underlying condition is F = −kx: the restoring force is directly proportional to displacement and directed towards equilibrium — this identifies which physical systems actually undergo SHM.

Worked example. A 0.60 kg mass oscillates on a spring of constant 15 N/m, amplitude 0.080 m. T = 2π√(m/k) = 2π√(0.60/15) = 2π√0.04 = 1.26 s. ω = 2π/T = 5.0 rad/s. Maximum acceleration (at x = A): a_max = ω²A = 25 × 0.080 = 2.0 m/s².

Standard systems:

pendulum:   T = 2 pi sqrt(L/g)        (small angles only)
mass-spring: T = 2 pi sqrt(m/k)

Neither period depends on amplitude — that is what makes them useful as timekeepers, and it is a frequent question.

Phase relationships

Displacement, velocity and acceleration are all sinusoidal but out of step:

• Velocity leads displacement by 90° — velocity is maximum at zero displacement, and zero at maximum displacement.
• Acceleration is 180° out of phase with displacement — maximum and opposite at the extremes, zero at the centre.

So at the equilibrium position: displacement zero, velocity maximum, acceleration zero. At maximum displacement: velocity zero, acceleration maximum. Reading those two states correctly answers most graph questions.

Energy

Background — beyond the specification, not among outcomes 143-145 or the supplied formulae (and the potential-energy form below is, in any case, only an approximation for the named simple pendulum):

Ek = 1/2 m omega^2 (A^2 - x^2)
Ep = 1/2 m omega^2 x^2
E_total = 1/2 m omega^2 A^2      constant

Kinetic energy is maximum at the centre; potential energy is maximum at the extremes; the total is constant in the absence of damping.

Both energy curves have twice the frequency of the displacement curve, because energy is maximum twice per cycle.

Damping

Resistive forces remove energy, so amplitude decreases.

Type	Behaviour
Light	Amplitude decays gradually; many oscillations
Heavy	Slow return to equilibrium without oscillating
Critical	Fastest return to equilibrium without overshooting

Critical damping is the fastest return, not the slowest. It is used in car suspension and measuring instruments, precisely because overshoot and oscillation are undesirable.

Damping reduces amplitude but has only a small effect on the natural frequency in the lightly damped case. Plastic deformation of ductile materials within a real oscillating structure also dissipates energy, adding to damping’s effect in reducing amplitude over successive cycles.

Resonance

A free oscillation happens at a system’s own natural frequency with no external driving force; a forced oscillation is driven by an external periodic force, potentially at a different frequency. Resonance occurs when the driving frequency equals the natural frequency, giving maximum amplitude and maximum energy transfer. CORE PRACTICAL 16 uses this relationship in reverse — determining an unknown mass from the resonant frequencies of oscillation of known masses.

Increasing damping reduces the peak amplitude and makes the resonance curve broader, and shifts the peak slightly to a lower frequency.

Examples: a swing pushed in time; a wine glass shattered by sound; MRI; radio tuning; and the destructive cases — bridges and buildings in earthquakes, which is why dampers are engineered in.

Exam traps

• Omitting the minus sign or the “towards equilibrium” clause from the SHM definition.
• Saying pendulum period depends on amplitude or on mass.
• Saying acceleration is maximum at the centre — it is zero there.
• Saying critical damping is the slowest return.
• Forgetting that the pendulum formula requires small angles.
• Using degrees where radians are needed for ω.

Self-test

1. Define simple harmonic motion fully.
2. Where in the cycle are velocity and acceleration each maximum?
3. What does the period of a pendulum depend on, and what does it not?
4. Distinguish light, heavy and critical damping.
5. What happens to a resonance curve as damping increases?
6. State the condition, in terms of force, for a system to undergo SHM.
7. A 0.60 kg mass oscillates on a spring of constant 15 N/m with amplitude 0.080 m. Calculate the period and the maximum acceleration.
8. Distinguish a free oscillation from a forced oscillation.

Answers: 1. Acceleration is proportional to displacement from equilibrium and is always directed towards the equilibrium position. 2. Velocity is maximum at the equilibrium position where displacement is zero; acceleration is maximum at maximum displacement. 3. It depends on length and gravitational field strength; it does not depend on amplitude (for small angles) or on the mass of the bob. 4. Light damping gives a slowly decaying oscillation; heavy damping returns to equilibrium slowly without oscillating; critical damping returns in the shortest time without overshooting. 5. The peak amplitude falls, the curve becomes broader, and the peak shifts slightly to a lower frequency. 6. F = −kx — the restoring force is directly proportional to displacement and directed towards equilibrium. 7. T = 2π√(0.60/15) = 1.26 s; ω = 2π/T = 5.0 rad/s; a_max = ω²A = 25 × 0.080 = 2.0 m/s². 8. A free oscillation occurs at the system’s own natural frequency with no external driving force; a forced oscillation is driven by an external periodic force, possibly at a different frequency.