Waves and Quanta PA114 Unit 1: Oscillations and Oscillators (Introduction) Tipler, Chapter 14 www.astro.le.ac.uk/~rda5/PA1140 Dr Richard Alexander (G44B)
Oscillators Pendulum Mass on a spring Tuning circuit Atomic bond fork Quartz crystal
Introductory lecture - Simple harmonic motion (SHM) - Angular frequency, phase, and amplitude - Energy in SHM - Damping, forcing - Resonance
Anharmonic oscillator E U = mgh Total Energy E = K + U h x
Simple Harmonic Motion oscillator Displacement x-direction spring constant x < 0 x > 0 Restoring force: Whether the spring is stretched or compressed, the restoring force acts towards the equilibrium position and is linearly related to the displacement (Hooke's Law): Simple Harmonic Motion
T
A - amplitude (maximum displacement) T0 - natural period (duration of cycle) f0 - frequency (no. of cycles per second or Hz) w0 - angular frequency (no. of radians per second) T
A - amplitude (maximum displacement) T0 - natural period (duration of cycle) f0 - frequency (no. of cycles per second or Hz) w0 - angular frequency (no. of radians per second) T
A - amplitude (maximum displacement) T0 - natural period (duration of cycle) f0 - frequency (no. of cycles per second or Hz) w0 - angular frequency (no. of radians per second) T
A - amplitude (maximum displacement) T0 - natural period (duration of cycle) f0 - frequency (no. of cycles per second or Hz) w0 - angular frequency (no. of radians per second) T
T
A - unconstrained, d - unconstrained Newton’s 2nd Law: Solution: Parameters: A - unconstrained, d - unconstrained
SHM and circular motion Displacement Initial phase Phase
What is the energy in the system? Energy is put into the system by the initial compression or stretching of the spring (work done = potential energy) The system also has kinetic energy associated with the motion of the mass
E P.E. = 1/2 kx2 T.E. = 1/2 kA2 A K.E. = 1/2 mv2 x stretch compress
E x Period the same Velocities lower A P.E. = 1/2 kx2 T.E. = 1/2 kA2 stretch compress
Physics is about looking for patterns
Oscillators Pendulum Mass on a spring Tuning circuit Atomic bond fork Quartz crystal
Oscillators store energy - like a battery or reservoir Oscillators measure time - unlike a battery or reservoir
explains temperature, thermal expansion, melting, ... Atomic bonds are “springy” T.E. explains temperature, thermal expansion, melting, ...
Damped SHM v friction frictional force: -v damping constant
Damped SHM
Forced, damped SHM: Resonance
Forced, damped SHM: Resonance Tidal resonance
Forced, damped SHM: Resonance Orbital resonance
Forced, damped SHM driving frequency Oscillations at driving frequency, w ; amplitude depends on how close w is to w0.
Coupled oscillators