Transcript A Brief History of Planetary Science

Damped and Forced SHM
Physics 202
Professor Lee Carkner
Lecture 5
If the amplitude of a linear oscillator is
doubled, what happens to the period?
a)
b)
c)
d)
e)
Quartered
Halved
Stays the same
Doubled
Quadrupled
If the amplitude of a linear oscillator is
doubled, what happens to the spring
constant?
a)
b)
c)
d)
e)
Quartered
Halved
Stays the same
Doubled
Quadrupled
If the amplitude of a linear oscillator is
doubled, what happens to the total
energy?
a)
b)
c)
d)
e)
Quartered
Halved
Stays the same
Doubled
Quadrupled
If the amplitude of a linear oscillator is
doubled, what happens to the maximum
velocity?
a)
b)
c)
d)
e)
Quartered
Halved
Stays the same
Doubled
Quadrupled
If the amplitude of a linear oscillator is
doubled, what happens to the maximum
acceleration?
a)
b)
c)
d)
e)
Quartered
Halved
Stays the same
Doubled
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) Increase
b) Decrease
c) 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) Increase
b) Decrease
c) 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) Increase
b) Decrease
c) 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 up
b) Move weight down
c) 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(wt)
k = mw2
w = (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
y-axis
Particle moving
in circle
of radius xm
viewed edge-on:
Particle at time t
xm
angle = wt+f
x-axis
cos (wt+f)=x/xm
x=xm cos (wt+f)
x(t)=xm cos (wt+f)
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=-wxmsin(wt+f)) from which we can
get the period
Vplanet
Center
of Mass
Star
Planet
Vstar
Orbits of a Star+Planet System
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
Fd = -bv

 Then,
x = xmcos(wt+f) 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:
w’ = [(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
wd = the frequency of the driving force
The amplitude of the motion will
increase the fastest when w=wd
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(wt+f)
v=-wxmsin(wt+f)
a=-w2xmcos(wt+f)
w=2p/T=2pf
F=-kx
w=(k/m)½ T=2p(m/k)½
U=½kx2
K=½mv2 E=U+K=½kxm2
Summary: Types of SHM
Mass-spring
T=2p(m/k)½
Simple Pendulum
T=2p(L/g)½
Physical Pendulum
T=2p(I/mgh)½
Torsion Pendulum
T=2p(I/k)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 w=wd