Damped and Forced SHM

Physics 202Professor Lee

CarknerLecture 5

If the amplitude of a linear oscillator is doubled, what happens to the period?

a) Quarteredb) Halvedc) Stays the samed) Doublede) Quadrupled

If the amplitude of a linear oscillator is doubled, what happens to the spring constant?

a) Quarteredb) Halvedc) Stays the samed) Doublede) Quadrupled

If the amplitude of a linear oscillator is doubled, what happens to the total energy?

a) Quarteredb) Halvedc) Stays the samed) Doublede) Quadrupled

If the amplitude of a linear oscillator is doubled, what happens to the maximum velocity?

a) Quarteredb) Halvedc) Stays the samed) Doublede) Quadrupled

If the amplitude of a linear oscillator is doubled, what happens to the maximum acceleration?

a) Quarteredb) Halvedc) Stays the samed) Doublede) Quadrupled

If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the period?

a) Increaseb) Decreasec) Stays the same

If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the maximum velocity?

a) Increaseb) Decreasec) Stays the same

If you have a pendulum of fixed mass and length and you increase the length of the path the mass travels, what happens to the maximum acceleration?

a) Increaseb) Decreasec) Stays the same

The pendulum for a clock has a weight that can be adjusted up or down on the pendulum shaft. If your clock runs slow, what should you do?

a) Move weight upb) Move weight downc) You can’t fix the clock by moving

the weight

PAL #4 Pendulums The initial kinetic energy is just the kinetic

energy of the bullet ½mv2 = (0.5)(0.01 kg)(500 m/s)2 =

The initial velocity of the block comes from the kinetic energy KE = ½mv2

v = (2KE/m)½ = ([(2)(1250)]/(5))½ = Amplitude =xm, can get from total energy

Initial KE = max KE = total E = ½kxm

xm =(2E/k)½ = ([(2)(1250)]/(5000))½ = Equation of motion = x(t) = xmcos(t)

k = m2

= (k/m)½ = [(5000/(5)]½ = 31.6 rad/s

Uniform Circular Motion

Simple harmonic motion is uniform circular motion seen edge on

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 angle =t+

Particle movingin circle of radius xm

viewed edge-on:

cos (t+)=x/xm

x=xm cos (t+) 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 byFd = -bv

Then,

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

e(-bt/2m) is called the damping factor and tells you by what factor the amplitude has dropped for a given time or:

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

Example:

Energy dissipation Example:

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 frequency of the driving force The amplitude of the motion will

increase the fastest when =d

Resonance

Resonance occurs when you apply maximum driving force at the point where the system is experiencing maximum natural force Example: pushing a swing when it is all the

way up All structures have natural frequencies

Next Time

Read: 16.1-16.5 Homework: Ch 15, P: 95, Ch 16, P:

1, 2, 6

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