oscillations, continued lecture 31 wednesday, november 19

Oscillations, continued

Lecture 31Wednesday, November 19

Workbook problems due Today

• WB 10.5, problems 14-25

Workbook Problems due Friday

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

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.

Equilibrium and Oscillation

• Frequency and Period

1

2

fTf

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

θ

w cos θ

w

w sin θ

T

,tangential sinnetF mg mg

Simple Pendulum

Find the angular frequency is

1 1

2 2

g

L

LT

g

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

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

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?

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) = ?

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

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