Oscillations (A Level)
17. Oscillations — Cambridge AS & A Level Physics 9702 Revision Notes & Topical Questions
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17. Oscillations — Cambridge AS & A Level Physics 9702 — CAMBRIDGE AS & A LEVEL PHYSICS 9702 — COMPLETE NOTES notes: Describe oscillations and simple harmonic motion, use the SHM equations and graphs, explain energy changes, and distinguish damping types and resonance.Oscillation…
- Equilibrium position
- Displacement x
- Amplitude x0
- Frequency f
- Angular frequency ω
- Phase difference
Section 1 of 9Describing an oscillation
- Oscillation = the repetitive variation with time of the displacement of an object about its equilibrium position (x = 0).
- Equilibrium position
- the position where no resultant force acts on the object.
- Displacement x
- the distance from equilibrium; a vector that can be positive or negative, in metres.
- Amplitude x0
- the maximum displacement on either side of equilibrium, measured from the equilibrium line to a peak.
- Period T
- the time for one complete oscillation, in seconds; SHM has a constant period.
- Frequency f
- the number of oscillations per second, in hertz, f = 1/T.What the symbols mean: T = period, the time for one complete wave (s) · f = frequency (Hz)
- the number of oscillations per second, in hertz, f = 1/T.
- Angular frequency ω
- ω = 2π/T = 2πf, in rad s−1.What the symbols mean: ω = angular frequency (rad s−1) · f = frequency (Hz) · T = period (s)
- ω = 2π/T = 2πf, in rad s−1.
Section 2 of 9Phase difference
- Phase difference = how far one oscillation is ahead of or behind another of the same frequency; a full cycle is 360° or 2π rad.
- In phase
- crests (or troughs) line up, with phase difference 0 or 360° (2π rad).
- Antiphase
- a crest lines up with a trough, with phase difference 180° (π rad).
- Quarter cycle
- a lead of ¼ of a cycle is 90° (π/2 rad), and the leading oscillation reaches each peak first.
Section 3 of 9Conditions for simple harmonic motion
- Simple harmonic motion (SHM) = oscillation in which the acceleration is proportional to the displacement from a fixed point and always directed towards it, a ∝ −x.
- There is a central equilibrium point.
- A restoring force acts towards it, with a magnitude proportional to the displacement.
- The restoring force gives the acceleration, so a and x always point in opposite directions.
- Examples
- a clock pendulum, a swing, a mass on a spring, guitar strings, a ruler vibrating off a table edge, a marble in a bowl and electrons in alternating current.
- Not SHM
- a person on a trampoline, since in the air the restoring force is the constant weight, which is not proportional to displacement.
- Forces
- a restoring force returns the object to equilibrium, while a resistive force opposes motion and causes damping.
Section 4 of 9Acceleration and displacement in SHM
- Acceleration
- a = −ω2x.What the symbols mean: a = acceleration (m s−2) · ω = angular frequency (rad s−1) · x = displacement from equilibrium (m); the minus sign means a always points back to the centre
- The magnitude of a is greatest at maximum displacement (x = ±x0).
- The minus sign means a is opposite to x.
- The a–x graph is a straight line through the origin with gradient −ω2.
- a = −ω2x.
- Displacement
- if the motion starts at x = 0 when t = 0, then x = x0 sin(ωt).
- The displacement equals x0 when sin ωt = ±1.
- The calculator must be in radian mode, and the sign of a negative displacement is kept.
- Starting at the amplitude
- the displacement is a cosine, x = x0 cos ωt.
Section 5 of 9Speed in SHM
- Velocity
- starting from equilibrium, v = v0 cos(ωt).
- Maximum speed
- v0 = ωx0, reached at the equilibrium position where x = 0.
- Speed against displacement
- v = ±ω√(x02 − x2); the speed is zero at the amplitude and greatest at the centre.
- Amplitude
- a larger amplitude means a larger distance in the same time, so a larger maximum speed.
- Conversions
- T, f and ω must be converted using ω = 2π/T = 2πf.
Section 6 of 9SHM graphs against time
| Graph | Shape (starting at equilibrium) | Link to the previous graph |
|---|---|---|
| Displacement | Sine curve | Reference |
| Velocity | Cosine curve | Gradient of the displacement graph, 90° out of phase |
| Acceleration | Negative sine curve | Gradient of the velocity graph, 90° out of phase, a mirror image of the displacement graph |
- Velocity
- it is maximum when the displacement is zero.
- Acceleration
- it is maximum when the displacement is maximum.
- Other starting points
- the curves shift but remain sines or cosines for undamped SHM.
Section 7 of 9Energy in SHM
- Energy interchange
- SHM swaps energy between kinetic and potential forms.
- A pendulum swaps gravitational potential and kinetic energy.
- A horizontal mass on a spring swaps elastic potential and kinetic energy.
- A vertical mass on a spring involves elastic potential, gravitational potential and kinetic energy.
- At the amplitude
- potential energy is maximum and kinetic energy is zero.
- At equilibrium
- kinetic energy is maximum and potential energy is minimum.
- Total energy
- it is constant, E = Ep + Ek = ½mω2x02.
- Energy–displacement graphs
- potential energy is a "U"-shaped curve, zero at x = 0 and maximum at ±x0.
- Kinetic energy is an inverted "U" ("n" shape), maximum at x = 0 and zero at ±x0.
- The total energy is a horizontal line above both curves.
- Per period
- two complete exchanges occur, because the amplitude is reached twice.
Section 8 of 9Damping
- Damping = the reduction in energy and amplitude of oscillations due to resistive forces such as friction or air resistance.
- It continues until the oscillator comes to rest at the equilibrium position.
- Frequency unchanged
- the frequency and period of a damped oscillator stay the same while the amplitude falls.
| Type | Behaviour | Example |
|---|---|---|
| Light damping | Oscillations continue with amplitude decaying exponentially | Pendulum gradually coming to rest |
| Critical damping | No oscillation; returns to equilibrium in the shortest possible time, with displacement falling quickly to 0 | Car suspension, weighing-scale needle |
| Heavy damping | No oscillation; returns to equilibrium very slowly without passing 0 | Door damper stopping slamming |
Section 9 of 9Forced oscillations and resonance
- Natural frequency f0
- the frequency at which a system oscillates freely.
- Driving frequency f
- the frequency of the periodic external force.
- Resonance = the maximum-amplitude response that occurs when the driving frequency equals the natural frequency.
- The driver supplies energy most efficiently, so the system gains maximum energy.
- If f is close to f0, the amplitude rises but less than at resonance.
- A child on a swing goes higher with small pushes timed to the swing's own frequency.
- Damping and resonance
- more damping lowers and broadens the resonance peak.
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