Topic 5 of 8 · Form 6
Circular motion, gravitation and oscillations
Angular displacement in radians; ω = 2π/T; v = rω; centripetal acceleration a = rω² = v²/r; centripetal force F = mv²/r.
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Circular motion
Angular displacement in radians; ω = 2π/T; v = rω; centripetal acceleration a = rω² = v²/r; centripetal force F = mv²/r.
Gravitational fields
F = GMm/r²; g = GM/r²; gravitational potential φ = −GM/r; orbits (GM/r² = v²/r); geostationary orbits (period 24 h, equatorial, same direction as Earth's rotation).
Worked example: Find the radius of a geostationary orbit (M = 6.0 × 10²⁴ kg; G = 6.67 × 10⁻¹¹ N m² kg⁻²; T = 86 400 s).
r³ = GMT²/(4π²) = (6.67 × 10⁻¹¹ × 6.0 × 10²⁴ × 86 400²)/(4π²) → r = 4.2 × 10⁷ m
Oscillations
Simple harmonic motion a = −ω²x; x = x₀ sin ωt; v = ±ω√(x₀² − x²); energy changes in SHM; damping (light, critical, heavy); forced oscillations and resonance.
Worked example: A mass oscillates with amplitude 0.05 m and frequency 2.0 Hz. Find its maximum speed.
ω = 2π × 2 = 12.6 rad s⁻¹ → vmax = ωx₀ = 0.63 m s⁻¹
Check yourself
- Why is gravitational potential always negative?
- Give one useful and one harmful example of resonance.
- What happens to the period of SHM as amplitude changes?
Show answers
- It is zero at infinity and work is done by the field as masses move together, so values are negative.
- Useful: microwave heating/tuning a radio; harmful: bridge or machine vibrations.
- It stays the same (independent of amplitude).
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