Download - Damped and Forced SHM
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