chapter 14 oscillations
TRANSCRIPT
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
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)(1N/m)(-.2m)2 =
(1/2)(1N/m)(.4m)2
next
back
Physlet E16.1 period vs. amplitude (spring and pendulum)Physlet E16.3 position and velocityPhyslet E16.6 under, critical, overdampedPhyslet E16.6 resonance (find f(resonant), m)
http://phet.colorado.edu/new/simulations/sims.php?sim=Masses_and_Springs
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1time
disp
lace
me
nt P
-1.5
-1
-0.5
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1time
disp
lace
me
nt P
At the point P, the mass has _______ and _______.
1) v>0, a>0 2) v=0, a>0 3) v<0, a>04) v>0, a=0 5) v=0, a=0 6) v<0, a=07) v>0, a<0 8) v=0, a<0 9) v<0, a<0
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?
1) 2)3)4)5)6)
tAe
BtAe
tAte
tAeBtAte
BAt
3dy
ydt
Which of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
2
22
d y dyy
dt dt
tAe
BtAe
tAte
tAeBtAte
BAt
Which of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
tdyy e
dt
tAe
BtAe
tAte
tAeBtAte
BAt
Which of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
2
2
d yy
dt
tAe
cos( )A tsin( )A tcosBtAe t
BAtsinBtAe t
5N/m
1kg
0.4m stretch
1N/m
1kg
0.5m stretch
5N/m
2kg
0.2m stretch
4N/m
5kg
0.2m stretch
4N/m
4kg
0.5m stretch
1N/m
5kg
0.5m stretch
Rank on the basis of time to complete one cycle. (Least to greatest)
A
B
C
D
E
F
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
B
C
D
E
F
6sin(3 )y t
3sin(6 )y t
6cos(3 )y t
6sin(3 30 )y t
10cos(6 )y t
10cos(2 )y t
Rank according to maximum velocity. (Least to greatest)
A
B
C
D
E
F
6cos(3 )y t
3cos(6 )y t
3cos(3 )y t
6cos(1.5 )y t
3cos(1.5 )y t
10cos(2 )y t
Rank according to maximum acceleration. (Least to greatest)
A
B
C
D
E
F
6cos(3 )y t
3cos(6 )y t
3cos(3 )y t
6cos(1.5 )y t
3cos(1.5 )y t
10cos(2 )y t
Which falls faster?
A: Meter stick B: Meter stick with heavy clamp
1) A2) B3) Same.4) 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) It increases.2) It stays the same.3) It decreases.4) More info is needed.
Which, if any, of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
2
2
d yy
dt
tAe
cos( )A tsin( )A t
cosBtAe t
BAtsinBtAe t
Which, if any, of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
tAe
BtAe
tAte
tAeBtAte
BAt
3dy
ydt
Which, if any, of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
2
22
d y dyy
dt dt
tAe
BtAe
tAte
tAeBtAte
BAt
Which, if any, of the following functions satisfy the given differential equation?
1) 2)3)4)5)6)
tdyy e
dt
tAe
BtAe
tAte
tAeBtAte
BAt