§ 17.1 Simple harmonic oscillations
Key ideas
- SHM is defined by a = −ω²x: the acceleration is proportional to the displacement and always directed back towards equilibrium.
- For a start at equilibrium, x = x₀ sin ωt; the velocity leads displacement by a quarter period and the acceleration is in antiphase with x.
- Speed is greatest at the centre and zero at the extremes: v = ±ω√(x₀² − x²), so v_max = ωx₀.
Equations
a = −ω² xthe defining condition of SHMm s⁻²
x = x₀ sin ωtdisplacement from the equilibrium startm
v = ±ω√(x₀² − x²)speed at displacement x; v_max = ωx₀m s⁻¹
Fig. 1 · The phase relations of SHM: velocity (dashed) peaks where displacement is zero, a quarter period ahead; acceleration (a = −ω²x) is the mirror image of the displacement curve.
Watch out: ω here is the angular frequency of the oscillation (ω = 2πf), not an angular speed around a circle. The minus sign in a = −ω²x is the whole physics: it is what makes the motion oscillate.
§ 17.2 Energy in simple harmonic motion
Key ideas
- Energy interchanges between kinetic and potential: all kinetic at the centre, all potential at the extremes.
- With no damping the total energy is constant and proportional to the square of the amplitude.
Equations
E = ½ m ω² x₀²total energy of the oscillationJ
Fig. 2 · Energy against displacement: KE (solid) is greatest at the centre, PE (dashed) at the extremes, and at every point they add to the same constant total.
Watch out: total energy goes as the square of the amplitude: doubling x₀ quadruples the energy stored, and likewise quadruples the maximum kinetic energy at the centre.
§ 17.3 Damped and forced oscillations, resonance
Key ideas
- Damping from resistive forces removes energy, shrinking the amplitude. Light damping decays slowly; critical damping returns to rest fastest without oscillating; heavy damping returns slowly without oscillating.
- A forced oscillation is driven at the driver's frequency. Resonance occurs when the driving frequency equals the system's natural frequency, giving maximum amplitude.
- More damping lowers and broadens the resonance peak and shifts it to a slightly lower frequency.
Fig. 3 · Resonance: amplitude peaks when the driving frequency matches the natural frequency f₀; heavier damping (dashed) gives a lower, broader peak.
Watch out: critical damping does not mean "no movement". It is the lightest damping that stops the system oscillating, returning it to equilibrium in the shortest time without overshoot.