oscillations, continued lecture 31 wednesday, november 19

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Oscillations, continued Lecture 31 Wednesday, November 19

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Page 1: Oscillations, continued Lecture 31 Wednesday, November 19

Oscillations, continued

Lecture 31Wednesday, November 19

Page 2: Oscillations, continued Lecture 31 Wednesday, November 19

Workbook problems due Today

• WB 10.5, problems 14-25

Page 3: Oscillations, continued Lecture 31 Wednesday, November 19

Workbook Problems due Friday

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

Page 4: Oscillations, continued Lecture 31 Wednesday, November 19

Quiz

• A student places her 500g physics textbook on a frictionless table. She pushes the book against a spring 4.00cm and then releases the book. What is the book’s speed as it slides away? The spring constant is k = 1250 N/m.

Page 5: Oscillations, continued Lecture 31 Wednesday, November 19

Equilibrium and Oscillation

• Frequency and Period

1

2

fTf

Page 6: Oscillations, continued Lecture 31 Wednesday, November 19

Simple Harmonic Motion

• Linear restoring force—– Example, mass on a spring

– Set y=0 at equilibrium point:–

,NET yF ky

k

m

2( ) cos( ) cos(2 ) cos

ty t A t A f A

T

Page 7: Oscillations, continued Lecture 31 Wednesday, November 19

θ

w cos θ

w

w sin θ

T

,tangential sinnetF mg mg

Page 8: Oscillations, continued Lecture 31 Wednesday, November 19

Simple Pendulum

Find the angular frequency is

1 1

2 2

g

L

LT

g

Page 9: Oscillations, continued Lecture 31 Wednesday, November 19

Description of motion

2 2

( ) cos( )

( )( ) sin( ) sin( )

( )( ) cos( ) ( )

MAX

x t A t

dx tv t A t v t

dtdv t

a t A t x tdt

Page 10: Oscillations, continued Lecture 31 Wednesday, November 19

0 1 2 3 4 5 6 7 8

-5

-4

-3

-2

-1

0

1

2

3

4

5

x(t) vs. t

x(t)

v(t)

a(t)

t sec

x(t

) m

ete

rs

Page 11: Oscillations, continued Lecture 31 Wednesday, November 19

0 1 2 3 4 5 6 7 8

-4

-3

-2

-1

0

1

2

3

4

x(t) vs. t

t sec

x(t)

met

ers

a) At what time(s) is particle moving right at maximum speed?

b) At what time(s) is particle moving right at maximum speed?

c) At what time(s) is the speed zero?

Page 12: Oscillations, continued Lecture 31 Wednesday, November 19

Problem 14:7

An air-track glider is attached to a spring. The glider is pulled to the right and released from rest at t=0s. It then oscillates with T=2.0s and vmax = 40cm/sa) A=?b) x(t=0.25s) = ?

Page 13: Oscillations, continued Lecture 31 Wednesday, November 19

Problem 14.72

2

.4m/s

.4A= 0.127m

MAX

T sT

v A A

( ) cos( ) 0.127cos( )

(.25) .127cos( (.25)) .0898

x t A t t

x

Page 14: Oscillations, continued Lecture 31 Wednesday, November 19

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

Page 15: Oscillations, continued Lecture 31 Wednesday, November 19

Pendulum1

2

g gf

L L

Point mass on a string

Page 16: Oscillations, continued Lecture 31 Wednesday, November 19

Physical Pendulum

θ

Center of gravityL

d

mgd

I

Page 17: Oscillations, continued Lecture 31 Wednesday, November 19

Damped Harmonic Motion

Friction rears its ugly head!

( )t

MAXx t Ae

Page 18: Oscillations, continued Lecture 31 Wednesday, November 19

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

Page 19: Oscillations, continued Lecture 31 Wednesday, November 19

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

Page 20: Oscillations, continued Lecture 31 Wednesday, November 19

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

Page 21: Oscillations, continued Lecture 31 Wednesday, November 19

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?

Page 22: Oscillations, continued Lecture 31 Wednesday, November 19

Problem 14.30

22 2 2 2

1

2(5 )(9.8)(.18)

.087 kg m4 4 (1.6)

mgdf

Imgd kg

If