LucidSTEM
A-LEVEL 9702 · A2 · TOPIC 17
Oscillations
The chain of the topic: one defining condition, a = −ω²x, fixes sinusoidal motion; from it follow the displacement and velocity solutions, then the smooth interchange of kinetic and potential energy, and finally how damping drains that energy and a driving force can feed it back at resonance. Around the hexagon are the ideas; above is what it builds on, below is where it leads.
TOPIC 17: OSCILLATIONS
CAMBRIDGE A-LEVEL PHYSICS 9702 · PATHWAYS
TheLucidSTEM · thelucidstem.com
BUILDS ON
T2 Kinematics: x, v, a T3 Newton's 2nd law: F = ma T12 Circular motion: ω = 2πf, projection 17.1
17.1
17.2
17.3
TOPIC 17
SIMPLE
HARMONIC MOTION
1 · DEFINING SHM
Acceleration toward equilibrium, set by displacement.
Condition: a ∝ x, and a is always opposite to x.
Amplitude x₀, period T, frequency f, ω = 2πf = 2π/T
Phase difference φ compares two oscillators in rad.
a = −ω² x
the defining equation of SHM
equilibrium
+x
a
when displaced +x, acceleration a points back to centre
2 · SOLUTIONS & GRAPHS
Sinusoids in time, a quarter cycle out of step.
Displacement: x = x₀ sin ωt (or x₀ cos ωt).
Velocity leads x by π/2; speed is greatest at x = 0.
Max speed v₀ = ωx₀; v = 0 at the turning points x = ±x₀.
v = v₀ cos ωt, v = ±ω√(x₀² − x²)
t
x
v
v (dashed) peaks a quarter period before x
3 · ENERGY IN SHM
Kinetic and potential trade, but the total is fixed.
KE is greatest at the centre; PE is greatest at ±x₀.
If undamped, total energy E stays constant.
KE = ½mω²(x₀² − x²), PE = ½mω²x²
E scales with the square of the amplitude.
E = ½ m ω² x₀²
x
0
−x₀
+x₀
E
PE
KE
KE + PE = E at every displacement
4 · DAMPING & RESONANCE
Resistive forces drain energy; driving can restore it.
Damping removes energy, so amplitude decays.
Light: many shrinking swings; critical: fastest return,
no overshoot; heavy: slow creep with no oscillation.
Resonance: driving at f₀ gives maximum amplitude.
More damping lowers and broadens the peak.
t
light damping: amplitude decays inside an envelope
LEADS TO
T21 Alternating currents: x = x₀ sin ωt for V and I in time T20 Magnetic fields: rotating coil drives a sinusoidal emf Each reuses one idea: a sinusoidal quantity x = x₀ sin ωt with angular frequency ω = 2πf, the signature of SHM.