Damped and Forced SHM Physics 202 Professor Lee Carkner Lecture 5

PAL #4 Pendulums  Double amplitude (x m )   k depends only on spring, stays same   v max = -  x m, increases   Increase path pendulum travels   v must increase (since T is constant, but path is longer) so max KE increases  If max KE increases, max PE increases  Clock runs slow, move mass up or down?   Since T = 2  (L/g) ½, want smaller L, move weight up

Uniform Circular Motion   Consider a particle moving in a circle with the origin at the center   The projection of the displacement, velocity and acceleration onto the edge-on circle are described by the SMH equations

UCM and SHM

Uniform Circular Motion and SHM x-axis y-axis xmxm  t+  Particle moving in circle of radius x m viewed edge-on: x(t)=x m cos (  t+  ) Particle at time t

Observing the Moons of Jupiter   He discovered the 4 inner moons of Jupiter   He (and we) saw the orbit edge on

Galileo’s Sketches

Apparent Motion of Callisto

Application: Planet Detection   The planet cannot be seen directly, but the velocity of the star can be measured   The plot of velocity versus time is a sine curve (v=-  x m sin(  t+  )) from which we can get the period

Orbits of a Star+Planet System Star Planet Center of Mass V star V planet

Light Curve of 51 Peg

Damped SHM  Consider a system of SHM where friction is present   The damping force is usually proportional to the velocity   If the damping force is represented by  Where b is the damping constant  Then, x = x m cos(  t+  ) e (-bt/2m)  x’ m = x m e (-bt/2m)

Energy and Frequency  The energy of the system is: E = ½kx m 2 e (-bt/m)   The period will change as well:  ’ = [(k/m) - (b 2 /4m 2 )] ½ 

Exponential Damping

Damped Systems   Most damping comes from 2 sources:  Air resistance   Energy dissipation   Lost energy usually goes into heat

Damping

Forced Oscillations  If you apply an additional force to a SHM system you create forced oscillations   If this force is applied periodically then you have 2 frequencies for the system  =  d =  The amplitude of the motion will increase the fastest when  =  d

Tacoma Narrows Disaster

Resonance    Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force   All structures have natural frequencies 

Summary: Simple Harmonic Motion x=x m cos(  t+  ) v=-  x m sin(  t+  ) a=-  2 x m cos(  t+  )  =2  /T=2  f F=-kx  =(k/m) ½ T=2  (m/k) ½ U=½kx 2 K=½mv 2 E=U+K=½kx m 2

Summary: Types of SHM  Mass-spring T=2  (m/k) ½  Simple Pendulum T=2  (L/g) ½  Physical Pendulum T=2  (I/mgh) ½  Torsion Pendulum T=2  (I/  ) ½

Summary: UCM, Damping and Resonance  A particle moving with uniform circular motion exhibits simple harmonic motion when viewed edge-on  The energy and amplitude of damped SHM falls off exponentially x = x undamped e (-bt/2m)  For driven oscillations resonance occurs when  =  d