Announcements
Chapter 13.1-13.3
December 1, 2009
Simple Harmonic Oscillators (SHOs)
The whole universe is composed of SHOs
1. If you have arranged to take the comprehensive make-up
on Dec. 9, send an email by Dec. 3 to “ilovephysics” with
the subject line “make-up exam” and list your excuse in
the body.
2. Last homework will be posted tomorrow—Due Tuesday
(Dec. 8) at 11:59 pm.
3. “Group 1” homework is a special test—not your
assignment.
4. Next Tuesday class will be review and evaluation of me.
There will be no clicker points next Tuesday for the
review.
Hooke’s Law Applied to a Spring – Mass System
What are they?
A system with energy sloshing between kinetic and potential
A system that can stretch and go back to original shape
A system with linear restoring force to original shape
Hooke’s Law gives the force F
=-kx
Acceleration of an Object in Simple Harmonic Motion
• Newton’s second law and Hooke’s Law
F=-kx=ma
a=-kx/m
• When x is positive (to the right),
F is negative (to the left)
• When x = 0 (at equilibrium), F
is 0
• When x is negative (to the left),
F is positive (to the right)
•The acceleration is a function of position
Acceleration is not constant
Uniform accelerated motion equation a No-No!
A C
B
A B C
Think about the position x,
speed v, and
acceleration a
of the mass as it moves through a cycle of amplitude A
Pendulums and Springs
Mass hanging on a Spring in Gravity
Frestoring = kx
2A
Frestoring = kx
sin θ ≈ θ for θ small
Frestoring = mgθ
linear in θ
x
-A
Mass pulls spring distance d
We pull mass a further distance A
and let go
Oscillates vertically-Becomes wave in time
A
1
Period and Frequency
x
Period = T = time of one cycle
= time to go around circle
at angular speed ω
v
2A
Frequency =
Spring Potential and Mass Kinetic Energy
1
1
PEs = kx 2
KE = mv 2
2
2
PEs + KE = constant
=
a
1 ω
=
T 2π
A 2 Joule Mass & Spring Oscillator
1 2 1 2 1
kx + mv = mvi 2
2
2
2
Solving v 2 = v 2 − k x 2
i
For v2
m
k
Note v = 0 when x = A or vi 2 = A2
m
v=
k 2
( A − x2 )
m
x=A
Chapter 13.4-13.7
The whole universe is composed of SHOs
SHO is system with linear restoring force to original shape/position
Note: For a mass/spring a = - k x / m not constant
True for all oscillators
Mass/spring
1 ω
Frequency f = =
T 2π
ωms
k
=
m
Pendulum
ωp =
g
L
2