Resonance

Lecture 32November 21, 2008

Robert Hooke

• “ceiiinosssttuv”• Anagram for “ut tensio, sic vis”• “as the extension, so the force”

Workbook Problems due Friday

• Problems 14-1 through 8, pages 14-1 -- 5

Energy in Simple Harmonic Motion2

2 2

2

2

1

21 1

constant2 2

1E(at x= A)=U

21

(at x=0)=K2

1

2

S

MAX

MAX MAX

MAX

U kx

E K U mv kx

kA

E mv

kv A

m

k kf

m m

Pendulum1

2

g gf

L L

Point mass on a string

Physical Pendulum

θ

Center of gravityL

d

mgd

I

Damped Harmonic Motion

Friction rears its ugly head!

( )t

MAXx t Ae

Damped Harmonic Motion

0 2 4 6 8 10 12

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Problem 14.15

A) When the displacement of a mass on a spring is ½A, what fraction of the mechanical energy is kinetic energy and what fraction is potential?

2 22

2

1 1 1

2 2 2 2 4

1

21

43

4

S

S

A AU kx k k

E kA

U

EK

E

B) At what displacement as a fraction of A, is the energy half kinetic and half potential?

2 22

2

2

1 1 1*

2 2 2

2 2

SU kx kA

A Ax

Problem 14.30The center of gravity of a lower leg of a cadaver which had a mass of 5kg was located 18cm from the knee. When pivoted at the knee , the oscillation frequency was 1.6Hz. What is the moment of inertia of the lower leg?

Problem 14.30

22 2 2 2

1

2(5 )(9.8)(.18)

.087 kg m4 4 (1.6)

mgdf

Imgd kg

If

Problem 14.33

• The amplitude of an oscillator decreases to 36.8% of it initial value in 10.0s. What is the time constant?

Problem 14.33

10.

( )

(0)

(10.0).368

(0)

10.ln(.368)

10.10.0

ln(.368)

t

MAX

MAX

MAX

MAX

x t Ae

x A

xe

x

s

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

x(t) vs. t

x(t)

met

ers

2 4 6 8

The period of this oscillator is approximately

The period of the oscillator is

1s 2s 5s 10s

25% 25%25%25%1. 1s2. 2s3. 5s4. 10s

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

x(t) vs. t

x(t)

met

ers

2 4 6 8

The is zero at t = ?? approximately

The velocity is zero when t =

1.25 s

2.60 s 5.2s

0.0 s

25% 25%25%25%1. 1.25 s2. 2.60 s3. 5.2s4. 0.0 s

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

x(t) vs. t

x(t)

met

ers

2 4 6 8

The acceleration is a maximum when t = ??

The acceleration is max when t=

0.00s

1.25 s 4.0 s

None of t

he above

25% 25%25%25%1. 0.00s2. 1.25 s3. 4.0 s4. None of the above

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

x(t) vs. t

x(t)

met

ers

2 4 6 8

The velocity is a maximum for t = ??

The velocity is a maximum for t =

0.0s 1.25s

2.6s 4.0s

25% 25%25%25%1. 0.0s2. 1.25s3. 2.6s4. 4.0s

Problem 14. 37

• A 25 kg child sits on a 2.0m long rope swing. To maximize the amplitude of the swinging, how much time should be between pushes?

Problem 14.37

2.02 2 2.84

9.8

LT s

g

Problem 14.32

A thin, circular hoop with a radius of 0.22m is hanging from its rim on a nail. When pulled to one side and released, the hoop swings back and forth. The moment of inertia for a hoop with the axis passing through the circumference is I = 2MR2. What is the period of oscillation?

Problem 14.32

2

2

2 22 2 1.33

IT

mgd

mR RT s

mgR g

Exam IV Wednesday, December 3

Chapter 10 and 14Quick Review Monday