17.1aDescribing oscillations
Definitions
- Displacement (x)
- the distance from the equilibrium position in a specified direction.
- Amplitude (x0)
- the maximum displacement from the equilibrium position.
- Period (T)
- the time taken for one complete oscillation.
- Frequency (f)
- the number of oscillations per unit time.
- Angular frequency (ω)
- the rate of change of phase angle; ω = 2πf.
- Phase difference
- the fraction of a cycle between two oscillations, expressed as an angle (2π rad = one cycle).
Key points
- One complete oscillation corresponds to a phase change of 2π rad (360°).
- Two oscillations with the same frequency and a time lag Δt have phase difference 2πΔt / T.
- In phase: difference 0 (or 2π); in antiphase: difference π rad.
- Angular frequency is measured in rad s-1; it is not an angular speed of anything physically rotating.
Displacement-time graph: amplitude x0 is the peak height; period T is one full cycle.
Oscillation B lags A by a quarter period: phase difference π/2 rad (90°).
Watch out!
- Phase difference in radians: use 2πΔt/T, not Δt/T.
- Amplitude is measured from the equilibrium position, not peak to trough.
17.1bSimple harmonic motion: definition and equations
Definitions
- Simple harmonic motion
- motion in which the acceleration is proportional to the displacement from a fixed point and is always directed towards that point.
Key points
- Defining equation a = -ω2x: the minus sign means acceleration is opposite to displacement.
- The restoring force is proportional to displacement and directed towards equilibrium.
- Period of SHM is independent of amplitude (isochronous).
- Starting at equilibrium: x = x0 sin ωt and v = v0 cos ωt. Starting at an extreme: x = x0 cos ωt.
- Maximum acceleration ω2x0 at the extremes; maximum speed ωx0 at the centre.
- A graph of a against x is a straight line through the origin with gradient -ω2.
a against x for SHM: straight line through the origin, gradient -ω2.
Mass-spring oscillator: displaced by x, the restoring force points back towards equilibrium.
Watch out!
- Forgetting the minus sign: a and x are always in opposite directions.
- Use radians on the calculator for sin ωt and cos ωt.
17.1cGraphs of displacement, velocity and acceleration
Definitions
- Antiphase
- a phase difference of π rad (half a cycle) between two oscillations.
Key points
- Velocity leads displacement by a quarter cycle (π/2 rad).
- Acceleration is in antiphase with displacement (π rad).
- v is the gradient of the x-t graph; a is the gradient of the v-t graph.
- At the extremes: x = ±x0, v = 0, |a| maximum. At the centre: x = 0, |v| maximum, a = 0.
- A v-x graph is an ellipse; its intercepts are ±x0 and ±v0.
x, v and a against time: the dashed line marks an extreme, where v = 0 and a is most negative.
v against x is an ellipse: v = 0 at x = ±x0, maximum speed v0 = ωx0 at x = 0.
Watch out!
- Where x is maximum, v is zero, not maximum.
- Velocity and acceleration are not in phase: a is greatest where v is zero.
17.2Energy in simple harmonic motion
Definitions
- Energy interchange in SHM
- kinetic energy and potential energy continuously interchange, the total energy remaining constant (no damping).
Key points
- At the centre: energy all kinetic. At the extremes: energy all potential.
- Total energy ∝ x02: doubling the amplitude quadruples the energy.
- Ek = ½mω2(x02 - x2); Ep = ½mω2x2; they add to the constant total.
- Against time, Ek and Ep each vary at twice the oscillation frequency.
- Energy graphs against x are parabolas; against t they are sin2 and cos2 shapes, never negative.
Energy against displacement: KE greatest at the centre, PE at the extremes; the sum is constant.
Energy against time (starting at the centre): KE and PE swap twice per period.
Watch out!
- Energy goes as amplitude squared: halving x0 gives a quarter of the energy.
- KE and PE vary with period T/2, not T.
17.3aDamped oscillations
Definitions
- Damping
- the loss of energy from an oscillating system, caused by resistive forces, resulting in a decrease in amplitude.
- Light damping
- the system oscillates with gradually decreasing amplitude.
- Critical damping
- the system returns to equilibrium in the shortest possible time without oscillating.
- Heavy damping
- the system returns slowly to equilibrium without oscillating.
Key points
- Resistive forces (friction, air resistance, viscosity) always act against the velocity, removing energy.
- Light damping: amplitude decays roughly exponentially; period almost unchanged.
- Critical damping is used in car suspensions and meter needles: fastest settling, no overshoot.
- Heavy damping: slower return than critical, still no oscillation.
Light damping: oscillations continue while the amplitude decays inside an exponential envelope.
Critical damping returns to equilibrium fastest with no overshoot; heavy damping creeps back slowly.
Watch out!
- Critical damping is the minimum damping that prevents oscillation, not "maximum damping".
- Damping reduces the amplitude, not (noticeably) the frequency, when it is light.
17.3bForced oscillations and resonance
Definitions
- Natural frequency (f0)
- the frequency at which a system oscillates when displaced and left to oscillate freely.
- Forced oscillation
- an oscillation in which a periodic driving force makes the system oscillate at the driving frequency.
- Resonance
- the maximum amplitude of vibration when the driving frequency equals the natural frequency of the system.
Key points
- A forced system oscillates at the driving frequency, not its natural frequency.
- At resonance energy is transferred from the driver most efficiently.
- Increasing damping: resonance peak lower and broader, and the peak moves to a slightly lower frequency.
- Useful resonance: microwave heating, MRI, tuning a radio circuit, musical instruments.
- Harmful resonance: bridges and buildings in wind or earthquakes, vibrating machinery; reduced by damping.
Resonance curves: amplitude peaks at the natural frequency f0; heavier damping gives a lower, broader peak.
Forced oscillation: a vibration generator drives a mass-spring system; vary the driving frequency to find resonance.
Watch out!
- Resonance is when the driving frequency equals the natural frequency; the system always oscillates at the driving frequency.
- Damping does not remove resonance; it lowers and broadens the peak.