Oscillations
Oscillations
Simple harmonic motion arises whenever a restoring force is proportional to displacement: F = −kx, giving a = −(k/m)x and the solution x = A cos(ωt + φ), with ω = sqrt(k/m) and T = 2π sqrt(m/k). A mass on an ideal spring and a small-angle pendulum (T = 2π sqrt(L/g)) are the standard models. In SHM, speed is maximum at equilibrium, acceleration is maximum at the turning points, and energy oscillates between ½kx² and ½mv² with the total ½kA² constant. The period is independent of amplitude for ideal SHM, and a pendulum's period is independent of its mass.
Traps: assuming the period depends on amplitude (it does only through the small-angle approximation, which fails for large swings); placing maximum acceleration at equilibrium rather than at the extremes; forgetting that ω in x = A cos ωt is an angular frequency, not a measured angular speed; and mishandling springs in series or parallel when asked for an effective k.
Worked example: a 0.5 kg mass on a spring with k = 200 N/m. Then ω = sqrt(200/0.5) = 20 rad/s and T = 2π/ω ≈ 0.31 s. With amplitude 0.05 m, the maximum speed is v_max = ωA = 1.0 m/s, reached at equilibrium.
Rule of thumb: write the equation of motion in the form a = −(constant)x; the square root of that constant is ω.