Chapter 14
Oscillations
www.youtube.com/watch?v=Rlk59xdM_YY
Introduction
• Oscillations of a Spring (Hands-on
emphasis)
• Simple Harmonic Motion (Mathematical
emphasis)
• Pendulums - Simple & beyond simple
• Damped Harmonic Motion (Modeling
emphasis)
• Driven Damped Harmonic Motion &
Resonance (the grand finale)
Oscillations of a Spring
• Characteristics
– Amplitude
– Period
– Frequency
– Phase
• Discovery Lab (Handout)
• Lab Project Assignment introduced
Simple Harmonic Motion
• Mathematical Representation
– Equation of motion (Simple common
phenomenon using Classical Mechanics)
– Solution exercise
– Role of initial conditions
– Phase angle
– Angular frequency and frequency
– Natural frequency
• Relation to Uniform Circular Motion
• Examples (Physlets)
Energy and SHM
• Kinetic energy of object in SHM
• Spring potential energy
• Potential energy graphical representation
– Whiteboard exercise
• Jeopardy problems 1 2 3 4 5
Pendulums
• Simple pendulum
– Equation of motion
– Approximation sin(θ) ≈ θ
• Handout or Exercise
– Solution
• Physical Pendulum
• Torsion Pendulum
Damped Harmonic Motion
• Equation of motion and solution
– Damping
– Over-damped, Under-damped, Critical
damping & Physlet
• Mathematical modeling
– Stella model (later)
Driven Damped Harmonic Motion
& Resonance
•
•
•
•
Driven (Forced) situations
Equation of motion and solution
Mathematical modeling continued
Resonance
– What? and When?
– Examples (including “field trip”)
– Q-value
the end
Is the function
Asin(ωt + ø) a solution of the general simple
harmonic motion equation?
If so, what are the constraints on ω, A and
ø?
back
To what question is this the
answer?
(1/2)(1kg)v2 = (1/2)(2N/m)(.2m)2
next
back
To what question is this the
answer?
(1/2)(1kg)v2 + (1/2)(1N/m)(-.2m)2 =
(1/2)(1N/m)(.4m)2
next
back
To what question is this the
answer?
(1/2)(3N/m)x2 = (1/2)(1kg)(1m/s)2
next
back
To what question is this the
answer?
(1/2)(2N/m)(.2m)2 = (1/2)(1N/m)x2
next
back
To what question is this the
answer?
(1/2)(1kg)(2m/s)2 = (1/2)k(2m)2
back
Physlet E16.1 period vs. amplitude (spring and pendulum)
Physlet E16.3 position and velocity
Physlet E16.6 under, critical, overdamped
Physlet E16.6 resonance (find f(resonant), m)
http://phet.colorado.edu/new/simulations/sims.php?sim=Masses_and_Springs
http://phet.colorado.edu/new/simulations/sims.php?sim=Masses_and_Springs
At the point P, the mass has _______ and
_______.
displacement
1.5
1
P
0.5
time
0
0
0.2
0.4
0.6
0.8
1
-0.5
-1
-1.5
1) v>0, a>0
4) v>0, a=0
7) v>0, a<0
2) v=0, a>0
5) v=0, a=0
8) v=0, a<0
3) v<0, a>0
6) v<0, a=0
9) v<0, a<0
Physlet E16.3 position and velocity
A mass oscillates on a spring. Consider two
possibilities: (i) v=0 and a=0 at some point in
time. (ii) v=0 at some point, but a≠0 at that
point. Which are true?
1)Both are.
2)Neither are.
3)Only (i)
4)Only (ii)
Which of the following functions satisfy the
given differential equation?
dy
 3 y
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Which of the following functions satisfy the
given differential equation?
d2y
dy
 2  y
2
dt
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Which of the following functions satisfy the
given differential equation?
dy
t
ye
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Which of the following functions satisfy the
given differential equation?
d2y
y
2
dt
1)
2)
3)
4)
5)
6)
t
Ae
A cos(t )
A sin(t )
Bt
Ae cos t
Ae Bt sin t
At B
Rank on the basis of time to complete one
cycle. (Least to greatest)
0.4m stretch
A
1kg
0.5m stretch
D
5N/m
1kg
1N/m
0.2m stretch
B
2kg
0.5m stretch
E
5N/m
4kg
4N/m
0.5m stretch
0.2m stretch
C
5kg
4N/m
F
5kg
1N/m
A mass is hanging in equilibrium via a spring.
When it is pulled down, what happens to the
total potential energy (gravity + spring)?
1)It increases.
2)It stays the same.
3)It decreases.
Rank on the basis of time to complete one
cycle. (Least to greatest)
A
y  6sin(3t )
B
y  3sin(6t )
C
y  6 cos(3t )
D
y  6sin(3t  30)
E
y  10 cos(6t )
F
y  10 cos(2t )
Rank according to maximum velocity. (Least
to greatest)
A
y  6 cos(3t )
B
y  3cos(6t )
C
y  3cos(3t )
D
y  6 cos(1.5t )
E
y  3cos(1.5t )
F
y  10 cos(2t )
Rank according to maximum acceleration.
(Least to greatest)
A
y  6 cos(3t )
B
y  3cos(6t )
C
y  3cos(3t )
D
y  6 cos(1.5t )
E
y  3cos(1.5t )
F
y  10 cos(2t )
Physlet E16.5,6 resonance
Physlet P16.3
Physlet P16.6
Which falls faster?
A: Meter stick
1)
2)
3)
4)
B: Meter stick with heavy clamp
A
B
Same.
More info is needed.
A pendulum is in an elevator that approaching the top
floor of a building and is coming to a stop. What
happens to the period of the pendulum?
1)
2)
3)
4)
It increases.
It stays the same.
It decreases.
More info is needed.
Which, if any, of the following functions satisfy
the given differential equation?
d2y
y
2
dt
1)
2)
3)
4)
5)
6)
A cos(t )
A sin(t )
t
Ae
Bt
Ae cos t
Ae Bt sin t
At B
Which, if any, of the following functions satisfy
the given differential equation?
dy
 3y
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Which, if any, of the following functions satisfy
the given differential equation?
d2y
dy
 2  y
2
dt
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Which, if any, of the following functions satisfy
the given differential equation?
dy
t
ye
dt
1)
2)
3)
4)
5)
6)
t
Ae
Ae  Bt
t
Ate
t
Ae
Ate
At B
Bt
Physlet 16.12 Floating oscillator