Chapter 15 - Oscillations

Simple Harmonic Motion (SHM) The red vector arrows are scaled to show the speed of the particle. The speed is maximum when the particle is at the origin and zero when it is at ±xm. Frequency (f) - # of oscillations per second (units of Hertz, abbreviated Hz) 1 Hertz = 1 Hz = 1 oscillation per second = 1 s-1 Period (T) – time for one complete oscillation, so f = 1/T and T = 1/f Animation here Movie here (15-2)

Understanding amplitude, period, and phase Different amplitudes only Different periods only Different phase only (try it)

Relating angular frequency, ω, to oscillatory frequency, f ω = radians per unit of time = radians in a full circle / period = 2Π/T and since 1/T = f ω = 2Πf (15-5)

Equations of SHM

Acceleration is proportional to displacement Combining Equations 15-3 and 15-7 yields the hallmark of simple harmonic motion a(t) = -ω2x(t) The acceleration is proportional to the displacement (15-8)

The Force Law for SHM (15-9) (15-12) (15-10) (15-13)

Energy in Simple Harmonic Motion Potential energy of an oscillator depends on the amount the spring is stretched or compressed (15-18) Kinetic energy of an oscillator depends on the speed of the block (15-19) Since (eqn. 15-12), we can write the above eqn. as (15-20)

Energy in Simple Harmonic Motion The total mechanical energy then is the sum of the potential and the kinetic energies E = U + K + Equals 1 (15-21)

Graphing U and K as a function of time and of position (a) Potential energy U(t), Kinetic energy K(t), and mechanical energy E as functions of time t for a linear harmonic oscillator. Note that all energies are positive and that U and K peak twice during each period. (b) Potential energy U(x), Kinetic energy K(x), and mechanical energy E as functions of position x for a linear harmonic oscillator with amplitude xm. For x=0 the energy is all kinetic and for x=±xm it is all potential.

Angular Simple Harmonic Oscillator Similar to Hooke’s Law, replacing variables with their angular equivalents… Torque kappa (torsion constant) angular displacement (15-22) Period Rotational Inertia kappa (torsion constant) Likewise, becomes (15-23)

The Pendulum (15-28) (15-29) T = period I = rotational inertia L = length of pendulum m = mass (kg) g = gravity h = axis to COM