Chapter 14

Oscillationswww.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 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

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

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?

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

Physlet E16.5,6 resonance

Physlet P16.3Physlet P16.6

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

Physlet 16.12 Floating oscillator