Ch

ap. 1

4: O

scillatio

ns,

Perio

dic M

otio

n,

Sim

ple H

arm

on

ic Motio

n

11

Characterized by:

Period (T) and Frequency (f)

Preparation for: M

echanical wave m

otionE

lectromagnetic w

ave

Dynam

ics: F= m

a, �= I �

�Equation of M

otion: G

eneral solution:

LLook

ing B

ack

to C

hap. 7

•

To describe oscillations in terms of

amplitude,

period, frequency

and angular frequency

•To

do calculations

with sim

ple harm

onic motion

(SHM

); To

analyze sim

ple harmonic m

otion using energy

•To apply the ideas of sim

ple harmonic

motion to different physical situations

•

To analyze

the m

otion of

a sim

ple pendulum

, followed

by a

physical pendulum

•To explore how oscillations die out

•To learn how a driving force can cause resonance

2

LList o

f Previo

usly C

overed

Topics

•To describe oscillations in term

s of am

plitude, period,

frequency and

angular frequency

•To

do calculations

with sim

ple harm

onic motion

(SHM

); To

analyze sim

ple harmonic m

otion using energy

•To apply the ideas of sim

ple harmonic

motion to different physical situations

•

To analyze

the m

otion of

a sim

ple pendulum

, followed

by a

physical pendulum

3

Calculus Trig. functions F = m

a (Chap. 4) Circular m

otion (Chap. 5) Restoring force (i.e. spring

force) (Chao. 6) Energy conservation (Chap. 7) M

oment of inertia (Chap. 9)

� = I ��(Chap. 10)�

•To

explore how

oscillations die out •

To learn

how a

driving force

can cause resonance

Oscillations

WWh

at is S

.H.M

.?

5

Periodic Motion

Horizontal oscillation

Vertical oscillation Vibration

Common characteristics

�Sim

plified Model

Simple H

armonic M

otion

Oscillations

SS.H

.M.

Usefu

l Math

an

d P

hysics

Trig. functions:

sin(��+ ��2) = cos(�) sin(��+ ����) = �sin(�) cos(��� ��2) = sin(�) D

erivative and integral T

rig. functions A

pproximation:

sin� ~ ��

S.H.O

. 1) Spring plus block�

Horizontal

Vertical 2) Pendulum

Sim

ple pendulum

Physical pendulum F = m

a

� = I ��

6

Oscillations

SS.H

.M.

Usefu

l Sk

ill –– Sk

etch /

Visu

aliza

tion

7

Oscillations

Circular M

otion to SHM

x R

0 ��

Starts here

8

x = R0 cos ��

where � = t

Oscillations

Analyzing Spring+B

lock System

10

Oscillations

S.H.M

. : Spring+Block system

and Sim

ple Pendulum system

Spring+block System

Simple Pendulum

12

Oscillations

Physical Pendulum – S.H

.M.?

Physical Pendulum

Simple Pendulum

m

Axis of rotation

15

Oscillations

��

��

Im

gdl g

xm k

a

�

�

�

Math about a and ��

XC

td

Xd

2

2

�

Oscillations

Physical Pendulum

Physical Pendulum – –

S.H.M

.?S.H

.M.?

Physical PendulumPhysical Pendulum

Simple Pendulum

Simple Pendulum

Oscillations

S.H.M

. : Spring+Block system

and S.H

.M. : Spring+B

lock system and

Simple Pendulum

systemSim

ple Pendulum system

Spring+block SystemSpring+block System

Simple Pendulum

Simple Pendulum

18

Acceleration is proportional to displacem

ent.

Oscillations

S.H.M

. – Dynam

ics (Summ

ary)

��

xC

td

xd

xC

Dt

CA

C

Dt

CA

CC

td

xd

Dt

CA

Cdt dx

Dt

CA

tx

Ct

d dx

Ct

dx

d

)cos(

)cos(

)sin(

:then

),

cos()

(

If

or

2

2 2

2

2

2

2

2

�

�

��

��

��

��

�

�� ��� �

�

�

��

Math about C

Physics about C

Oscillations

)cos(

)(

φωt

At

x��

22

0

Oscillations

Circular M

otion to SHM

x

x = R0 cos ���

where � = t

R0

x(t) = R0 cos ( t)

��

Starts here

Kin. Equation of SHM

x(t) = R0 cos (�������������)

��

21

Oscillations

Acceleration is proportional to displacem

ent. S.H

.M.

�

� �� � �� �

�

�

��

position)

starting

the

on

(depending

2)

cos(

)(

onaccelerati

2

2

� �

�

T

tA

tX

XC

Ct

dX

d

��

�

224

XC

td

Xd

2

2

�

Oscillations

WWh

at is � ?

26

t

x(t) �/

� = ��/2�

x(t) = Asin(

�t ) = A

cos( �t )

= A cos(

��t ���/) )

= A cos(

�t � � )

Oscillations

S.H.M

.

3 independent variables: A

ngular frequency��A

mplitude

Phase angle

x(t) = A cos(

�t + �)

�(t) = A cos(

�t + �)

Period: T = 1 / f �

T = 2�/�

Note: A

ngular frequency () is N

OT the angular velocity.

Exam

ple of x(t), w

here ��= 0

Quick C

heck: H

ow do v(t) and a(t) look like?

28

Oscillations

Graphs

330

Oscillations

3 31

Oscillations

3 32

Oscillations

3 33

Oscillations

(1)Identify the force along

the m

otion and find the

equation of motion.

(2) D

etermine the values of three independent variables

Am

plitude

A

ngular frequency (and period)

Phase angle

(3) Sketch ( x-t, vv-t, a-t ) or ( �-t, -t, �-t ) graphs

and find the m

aximum

values of …

Tech

nica

l Step

s

Im

gdI

mgd

ge

�

�

�

�

.,

.

XC

td

Xd

2

2

�

Oscillations

S.H.M

.S.H

.M.

3 independent variables:A

ngular frequencyA

mplitude

Phase angle

x(t) = Acos(

�t + �)

�(t) = Acos(

�t + �)

Period:T = 1 /f �

T= 2�/

Period:T = 1 /f �

T= 2�/

Note:A

ngular frequency () is N

OT

the angular velocity.

Exam

ple of x(t), w

here ��= 0

Quick C

heck:Q

uick Check:

How

do v(t) and a(t) look like?

35

Oscillations

Rotational M

otion

2(e) A solid disk (m

ass M= 3.00 kg and radius R

= 20.0 cm) is

hung from the w

all by means of a m

etal pin through the hole, and used as a pendulum

. Calculate the m

oment of inertia of the

disk about the pin (= the axis of the rotation).

Rotational M

otion

2(c) A m

eter stick (mass M

= 0.500 kg and length L= 1.00 m

) ishung from

the wall by m

eans of a metal pin through the hole,

and used as a pendulum. E

xpress the mom

ent of inertia of thestick about the pin (= the axis of the rotation) in term

s of M, L,

and x.

Rotational M

otion

Torque due to G

ravity?T

orque due to Gravity?

)(

sin2

�� !

"# $�

%

�

�

m

g

xl

F

r

�

��

r �x

F �

lC

Mm

assm

?

%

F

r

�

���

ations

Rotatitttittttittittiiiona

oooooooool M

otion

2(e) A solid

hung fromk (m

ass M= 0.500 kg and length L

= 1.00 m) is

all by means of a m

etal pin through the hole,d di

m the iske w

= 3.00ans of

M=

mea

ss Mby m

k (mas

wall b

kg f a m

Rrodius n th

and radm

etal pin

Looking B

ack at I and ���

36

Oscillations

Con

cepts o

f S.H

.M..

Dynam

ics & Kinematics

Spring+block �

F = m

a & (x, v, a)

Pendulum

� a part of circular m

otion���

� = I ���&

(�, , �) Force:

Conservative force

Restoring force

Conservation:�

K + U

= constant

+ S.H.M

. (�as angular frequency)

37

Oscillations E

xercise 1: Find �

= [Find I and d] 40

Step

1: P

hysica

l Pen

du

lum

s Im

gdI

mgd

�

�

�

�

Oscillations

4 43

Exercise 1: Find

� = [Find I and d]

Exercise 2: Find ���

= [ Find I, d, and ���]

I d

I, d��

�

Oscillations

Step

1: P

hysica

l Pen

du

lum

s (II)

Exercise 2: Find ���

= [ Find I, d, and ���] 44

Oscillations

4 45

Want to find T

Need to know �

Then, find I

Step

1: P

hysica

l Pen

du

lum

s (III)

OOOOOOOOOOOsssssssssssccccccccccciiiiiiiiiiillllllllllllllllllllllaaaaaaaaaaatttttttttttiiiiiiiiiiiooooooooooonnnnnnnnnnnsssssssssss

�Find I

�Calculate

�Then, find T

Exercise 3: Find T

Oscillations

Cau

tion

: An

gula

r Velo

city?

446

Oscillations

PPra

ctice Pro

blem

1

47

Oscillations

Exercise 1+: Find

�

��= [Find I and d]

Top-Dow

n Steps: 1.W

hat is asked?����

�

(rigid body) � I and d

2. How

to find I and d for a system of tw

o rigid bodies?

I = I1 + I2

3. How

to find I1 (or I2 )?

W

here is the c.m. position? U

se parallel-axis theorem

4. How

to find d ?

W

here is the c.m. position of the system

?

Im

gdI

mgd

�

�

�

�

48

Pra

ctice Pro

blem

2

kg 0

2 cm 0

5kg

500 cm

10

2 1

.m

.r

.m

�

Oscillations

Oscillations

x= 2.00 cm

x=

2.00cm

50

Oscillations

Nam

e: _______________ Section: ______ UIN

: __________

Work on (a)

Oscillaaa

Repeat (a) without and w

ith a particle (mass m

)

51

Oscillations

Nam

e: _______________ Section: ______ UIN

: __________

Oscillaaa

Work on (a), (b), (c), and (d) 55

2

Oscillations

Problem 4: (25 points)

Determ

ine the net torque (magnitude

and direction) due to gravity on the system about the

pin, shown in the figure below

. A beam

has mass M

and length l; a big solid sphere has m

ass Mand radius R; a sm

all sphere has mass M

/2 and radius R/2. Assum

e l> Rand l> 2x.

Also determ

ine the mom

ent of inertia of the system about the pin. [H

int: use Parallel-axis theorem

.]

l

ations

But, this can also be a C

hap.14 problem,

If I ask you to find . (angular frequency) in S.H

.M. of a physical pendulum

.

Chap. 10

53

Oscillations

AAn

ato

my o

f S.H

.M. P

roblem

+ S.H.M

. (�as angular frequency) 54

Chap. 9

Chap. 10

Chap. 8

Calculus

Oscillations

Example 8: Physical Pendulum

s�2.00 cm 55

Oscillations

Equation of Motion:

d2x/dt 2 +

2 x = 0 or d2�/dt 2 +

2 ��= 0 G

eneral solution:

x(t) = A cos( �t + �)

�(t) = A cos(�t + �) �

Position (x or �) as a function of time.

Am

plitude

Angular Frequency

Oscillations

Periodic Motion

Simple H

armonic M

otion (S.H

.M.)

56

Oscillations

Periodic Motion

T = 2�� L cos� / g T 2 = (4�

2/GM

) s 3

57

Oscillations

Example 1: (a) F

T = ? (b) T = ?

mg sin��

Be C

ritcal Thinker

gL

T�

�cos

2

g lT

�2

FT =m

gcos��F

T =mg/cos��

58

Oscillations

Torsion Pendulum

59

Oscillations

S.H.M

. : T = 2�/�

660

Oscillations

Example 2(A

)

+

+

+

+

+ 0

T/4

T/2

3T/4

T

Equilibrium

Positions

A

61

Oscillations

Example 2(B

)

Ea = E

b = Ec = E

d

62

Oscillations

Example 4: M

omentum

conservation!�

What is the speed of the bullet?

Express the speed in term

s of m, M

, k, and d.

d

664

Oscillations

Example 5

You holdthe block

at x = A (= 0.030 m

) by applying 6.0 N

. T

hen, the block was

released.

The m

otion of the block undergoes SHM

. C

an you show that a = –(k/m

) x ? A

lso find: (a) k

(b)�

(c) T

(d) vm

ax (where?, w

hen?)

(e) x, v and a at t = 2 sec

k 0.50 kg

66

Oscillations

Example 9: V

ertical S.H.M

.�

67

Oscillations

Physical Pendulum (I)

����

Small oscillation

�� = sm

all �

L|| = L cos� ~ L

L||

68

Oscillations

Physical Pendulum (II)

+ x

����

+

mg

x0 x = 0

Fsp = k x

0

Equilibrium

(&�i = 0) �

mg(L/2) – kx

0 (L) = 0

69

Oscillations

Physical Pendulum (III)

+ x

����

+

mg

x0 x = 0

Fsp = k x

Equation of m

otion (&�i = I �) &�i = m

g(L/2) – kx(L) = kx0 (L) – kx(L)

–k(x–x

0 )(L) = Irod(P) d2�/dt 2

P

+ x

L� = x– x

0

70

Oscillations

889