Damped and Forced SHM

Physics 202Professor Lee

CarknerLecture 5

PAL #4 Pendulums Double amplitude (xm)

k depends only on spring, stays same vmax = -xm, 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

xm

t+

Particle movingin circle of radius xm

viewed edge-on:

x(t)=xm 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=-xmsin(t+)) from which we can get the period

Orbits of a Star+Planet System

StarPlanet

Centerof Mass

Vstar

Vplanet

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 = xmcos(t+) e(-bt/2m)

x’m = xm e(-bt/2m)

Energy and Frequency The energy of the system is:

E = ½kxm2 e(-bt/m)

The period will change as well:’ = [(k/m) - (b2/4m2)]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=xmcos(t+)

v=-xmsin(t+)

a=-2xmcos(t+)

=2/T=2fF=-kx

=(k/m)½ T=2(m/k)½

U=½kx2 K=½mv2 E=U+K=½kxm2

Summary: Types of SHM

Mass-springT=2(m/k)½

Simple PendulumT=2(L/g)½

Physical PendulumT=2(I/mgh)½

Torsion PendulumT=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 = xundamped e(-bt/2m)

For driven oscillations resonance occurs when =d