Simple harmonic motion · Grade 12
A swinging pendulum, a bouncing mass on a spring and a vibrating guitar string all oscillate back and forth about a middle position. When the acceleration is proportional to the displacement and always points back to the middle, the motion is simple harmonic, and its period does not depend on the size of the swing.
Key facts
- Period T = 1/f; angular frequency ω = 2πf = (2π)/T.
- SHM: a = −ω²x (acceleration ∝ displacement, towards equilibrium).
- x = A cos ωt (released from the end); v_max = ωA at the centre; a_max = ω²A at the ends.
- Speed at displacement x: v = ω√(A² − x²).
- Mass on a spring: T = 2π√(m/k). Simple pendulum (small swings): T = 2π√(L/g).
- Total energy = ½kA² = ½mω²A². Damping removes energy; resonance happens when the driving frequency equals the natural frequency.