Oscillations
What this domain tests
Oscillations tests simple harmonic motion, its equations, energy, pendulums, springs, damping and resonance.
Core rules
- SHM: a = −ω²x; x = A sin(ωt + φ); v = ω√(A² − x²); v_max = Aω; a_max = Aω².
- T = 2π/ω; spring T = 2π√(m/k); simple pendulum T = 2π√(L/g).
- Energy: KE = ½mω²(A² − x²), PE = ½mω²x², total = ½mω²A².
- Damping reduces amplitude gradually; resonance occurs when driving frequency equals natural frequency.
- The phase difference between displacement and velocity is π/2.
Worked example
A mass of 0.25 kg on a spring of constant 100 N m⁻¹ oscillates. ω = √(k/m) = √(100/0.25) = 20 rad s⁻¹, so T = 2π/20 ≈ 0.314 s. If the amplitude is 0.05 m, maximum speed = Aω = 1.0 m s⁻¹ and maximum acceleration = Aω² = 20 m s⁻².
Traps
- Saying acceleration is in the direction of displacement; it is opposite.
- Thinking a pendulum's period depends on mass; for small swings it depends only on length and g.
- Mixing angular frequency ω with frequency f; ω = 2πf.
- Forgetting that total energy stays constant in undamped SHM.