Oscillations
Oscillations tests the kinematics and energetics of simple harmonic motion, the spring-mass and simple-pendulum systems, angular SHM, and the qualitative behaviour of damped and forced oscillations. Numerical questions on period, velocity and energy are standard.
Core rules
- Defining condition: a = −ω²x, with the acceleration always directed towards the mean position.
- Displacement x = A sin(ωt + φ); velocity v = Aω cos(ωt + φ); acceleration a = −Aω² sin(ωt + φ).
- At the mean position, speed is maximum (Aω) and acceleration is zero; at extremes, speed is zero and acceleration magnitude is Aω².
- Energy: KE = ½mω²(A² − x²), PE = ½mω²x², total E = ½mω²A², constant.
- Spring-mass: T = 2π√(m/k); springs in series 1/k = 1/k₁ + 1/k₂, in parallel k = k₁ + k₂.
- Simple pendulum: T = 2π√(L/g), valid for small angles. A physical pendulum uses T = 2π√(I/mgd).
- Angular SHM: α = −ω²θ, T = 2π√(I/C). A bar magnet oscillating in a field has T = 2π√(I/mB).
- Damped motion has amplitude falling as e^(−bt/2m); forced oscillation shows resonance when the driving frequency equals the natural frequency.
Worked example
A simple pendulum has L = 1.0 m and g = 9.8 m/s². Then T = 2π√(1.0/9.8) = 6.283 × 0.3194 = 2.01 s, so the frequency is about 0.50 Hz. Quadrupling the length would double the period, since T ∝ √L.
Traps
- Writing T = 2π√(k/m) for a spring - it is 2π√(m/k).
- Using the pendulum formula for large amplitudes, where the small-angle approximation fails.
- Claiming total energy varies in SHM; it stays constant as KE and PE exchange.