Inside This Unit: The Full Breakdown
This unit covers simple harmonic motion (springs and pendulums) and Newtonian gravitation, including orbits, gravitational potential energy U = −GMm/r, and escape velocity.
Why it matters
Oscillations and gravitation close the course, combining the restoring-force idea with energy methods and orbital mechanics.
Key concepts
- SHM has a restoring force F = −kx and position x(t) = A cos(ωt + φ).
- A spring oscillator has T = 2π√(m/k); a pendulum T = 2π√(L/g); period is independent of amplitude.
- Energy in SHM is ½kA², exchanging between KE and U.
- Gravitation is F = GMm/r² with U = −GMm/r; orbits use GMm/r² = mv²/r.
Simple Harmonic Motion
When the restoring force is proportional to displacement (F = −kx), motion is sinusoidal: x(t) = A cos(ωt + φ) with ω = √(k/m). Period depends on m and k (or L and g for a pendulum), not amplitude. Total energy ½kA² shifts between kinetic and potential, with maximum speed at equilibrium.
Gravitation and Orbits
Newton’s law of gravitation, F = GMm/r², is an inverse-square law with general potential energy U = −GMm/r (zero at infinity, not mgh). For circular orbits gravity provides the centripetal force, giving v = √(GM/r), and Kepler’s third law T² ∝ r³. Escape velocity follows from energy conservation: v = √(2GM/r).
AP exam tip
At astronomical scales use U = −GMm/r, not mgh — and remember orbital and escape-speed derivations come from setting gravity equal to the centripetal force or using energy conservation.
Connections to other units
- Unit 3: Energy conservation underpins both SHM and orbital mechanics.
- Unit 1: SHM uses the same x-v-a derivative relationships from kinematics.
