Oscillations - Quick Revision
SHM basics
- Periodic motion repeats in equal intervals; oscillatory motion is to-and-fro about a mean position.
- SHM: acceleration proportional to displacement and directed toward the mean position, a = -omega^2 x.
- x = A cos(omega t + phi); omega = 2 pi/T = 2 pi f.
Velocity, acceleration, force
- v = -A omega sin(...) = omega sqrt(A^2 - x^2); vmax = A omega at mean position.
- a = -omega^2 x; amax = omega^2 A at the extremes.
- Force law: F = -k x with omega = sqrt(k/m).
- SHM is the projection of uniform circular motion on a diameter.
Systems
- Spring-mass: T = 2 pi sqrt(m/k); springs in series/parallel change effective k.
- Simple pendulum: T = 2 pi sqrt(L/g) (small angles); independent of mass and amplitude.
Energy
- E = (1/2) k A^2 (constant); KE = (1/2)k(A^2 - x^2), PE = (1/2)k x^2.
- KE and PE each oscillate at twice the frequency; KE max at mean, PE max at extremes.
Damped and forced
- Damped: amplitude decays as e^(-bt/2m); energy lost to friction.
- Forced/resonance: amplitude is largest when the driving frequency equals the natural frequency.