Further Mechanics and Fields: Circular Motion, Simple Harmonic Motion, Gravitational and Electric Fields
Circular motion
Angular velocity ω = 2π/T = 2πf; speed v = ωr. Centripetal acceleration a = v²/r = ω²r, directed toward the centre. The resultant force must provide mv²/r; it is not an extra force. For a charged particle in a magnetic field, qvB = mv²/r gives r = mv/(qB).
Simple harmonic motion
SHM requires a ∝ −x. Then a = −ω²x, x = A cosωt and v = ±ω√(A² − x²). Maximum speed is ωA at the equilibrium position; maximum acceleration is ω²A at the extremes. For a mass–spring system T = 2π√(m/k); for a simple pendulum T = 2π√(l/g).
Gravitational and electric fields
g = GM/r² and V_g = −GM/r. For a circular orbit, GM/r² = v²/r, so v = √(GM/r) and T = 2πr/v. Escape speed is √(2GM/r). For a point charge, E = kQ/r² and V = kQ/r; a uniform field has E = V/d.
Worked example
A satellite orbits at r = 7.0 × 10^6 m from Earth's centre. Using GM = 3.99 × 10^14 m³ s^-2, v = √(3.99 × 10^14/7.0 × 10^6) = 7.55 × 10^3 m/s. Period T = 2πr/v = 2π × 7.0 × 10^6/7.55 × 10^3 = 5.8 × 10^3 s, about 97 minutes.
Traps
Adding a separate centripetal force; reversing the direction of SHM acceleration; using degrees in ωt; forgetting that gravitational potential is negative; using r as altitude rather than distance from the centre.