Oscillations

Monday, November 19Lecture 30

Workbook problems due Wednesday

• WB 10.5, problems 14-25

Workbook Problems due Friday

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

Power

• Power is the rate of transformation of energy

• Unit is 1 Watt=1W = 1 J/s• If energy being transformed is work, W then

EP

t

W F x xP F Fv

t t t

Is the work done by F + or - ?

Positive

negative

50%50%1. Positive2. negative

F

d

Is the work done by F + or - ?

Positive

negative

50%50%1. Positive2. negative

Fd

Is the work done by F + or - ?

Positive

negative

50%50%1. Positive2. negative

F

d

Problem 10:20

• A pendulum is made by tying a 500g ball to a 75-cm-long string. The pendulum is pulled 300 to one side and then released.

• A) What is the ball’s speed at the lowest point in its trajectory?

• B) To what angle does the pendulum swing on the other side.

Problem 10:20

• Use energy conservation

300 L=0.75 m

Δy=L-Lcos 300

Problem 10:20 cont

• Set y=0 at lowest point of swing

2

2

10 0

21

(1 cos )2

2 (1 cos ) 1.4 m/s

gi i gf fU K U K

mg y mv

mgL mv

v gL

Problem 10:24

• 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.

Problem 10:24

• 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.

Problem 10:24

• Using the initial position as the compressed spring, final after book leaves spring:

2 2

2

1 11250 / (.04 )

2 20 0

1

2

Si i sf f

Si i

i Sf

f f

U K U K

U kx N m m

K U

K mv

Problem 10:24

• Finally

2 21250 N/m(.0400cm)

0.500kg

2.00 m/s

f i

kv x

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

Simple Harmonic Motion

• If restoring force is linearly proportional to displacement (e.g. F=-kx) then we will have simple harmonic motion.

• In lab last week you experimented with a simple pendulum. Was its motion simple harmonic?

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

Wednesday

Oscillations continuedProblems CQ3,CQ9,MC18,MC19, 1, 4, 6, 7, 10