Oscillations An Introduction

Periodic Motion Repeats itself over a fixed and reproducible period of time. Mechanical devices that do this are known as oscillators.

An example of periodic motion… 3 2 4 6 t(s) -3 x(m)

Simple Harmonic Motion (SHM) Periodic motion which can be described by a sine or cosine function. Springs and pendulums are common examples of Simple Harmonic Oscillators (SHOs).

Equilibrium The midpoint of the oscillation of a simple harmonic oscillator. Position of minimum potential energy and maximum kinetic energy.

Law of Conservation of Energy All oscillators obey… Law of Conservation of Energy In pendulums, UG + K = Constant In springs, ½ kx2 + K = Constant

Amplitude (A) How far the wave is from equilibrium at its maximum displacement. Measured in meters (m) Waves with high amplitude have more energy than waves with low amplitude.

Period (T) The length of time it takes for one cycle of periodic motion to complete itself. Measured in seconds.

Frequency (f): How fast the oscillation is occurring. (oscillations/sec.) Frequency is inversely related to period. f = 1/T The units of frequency are Herz (Hz) where 1 Hz = 1 s-1.

Parts of a Wave T 3 A Equilibrium point 2 4 6 t(s) -3 x(m)

Restoring force The restoring force is the secret behind simple harmonic motion. The force is always directed so as to push or pull the system back to its equilibrium (normal rest) position.

x: displacement from equilibrium Hooke’s Law A restoring force directly proportional to displacement is responsible for the motion of a spring. F = -kx where F: restoring force k: force constant x: displacement from equilibrium

Hooke’s Law F = -kx F m x m F m x Equilibrium position F m Spring compressed, restoring force out x m Spring at equilibrium, restoring force zero F m Spring stretched, restoring force in x

Fs = -kx Fs mg Hooke’s Law m The force constant of a spring can be determined by attaching a weight and seeing how far it stretches. mg

Harmonic Motion Equations  = ωt = 2π/T ω = angular velocity = 2πf where f is frequency in Hz X = Acos() = Acos(ωt) = Acos(2πt/T) (X = position, A = amplitude, T = period)

Harmonic Motion Velocity & Acceleration v = -Aω sin(ωt) vMax = Aω a = -Aω2 cos(ωt) aMax = Aω2

Period of a spring T = 2m/k T: period (s) m: mass (kg) k: force constant (N/m)

Period of a pendulum T = 2l/g T: period (s) l: length of string (m) g: gravitational acceleration (m/s2)