Lecture

8(第八講)Rotation(Chapter

10)----轉(zhuǎn)動(dòng)ContentsThe

London

Eye(a

giant

Ferris

wheel)1.Motion

of

a

RigidBody(剛體的運(yùn)動(dòng))ties)Translational

motion

andRotationalmotion(平動(dòng)與轉(zhuǎn)動(dòng))Rotational

variables(or

Angular—有關(guān)“轉(zhuǎn)動(dòng)運(yùn)動(dòng)學(xué)”的四個(gè)變量The

rotational

motion

compares

to

thetranslational

motion(平動(dòng)與轉(zhuǎn)動(dòng)的對(duì)比)Kinematics(運(yùn)動(dòng)學(xué)的比較)Dynamics(動(dòng)力學(xué)的比較)Rotation

with

constant

angular

acceleration(勻角加速轉(zhuǎn)動(dòng))Relating

Linear

and

Angular

tiesUniversity

Physics

I1The

London

Eye(a

giant

Ferris

wheel)ContentsKinetic

Energy

of

Rotation(轉(zhuǎn)動(dòng)動(dòng)能)Rotational

Inertial

(Moment

of

Inertial)

&

????(轉(zhuǎn)動(dòng)慣量&轉(zhuǎn)動(dòng)動(dòng)能)Calculating

the

Rotational

Inertial(Moment

of

Inertial)轉(zhuǎn)動(dòng)慣量的計(jì)算Parallel-Axis

Theorem(平行軸定理)5.Newton’s

2nd

Law

for

Rotation

about

a

Fixed

Axis(剛體定軸轉(zhuǎn)動(dòng)的

第二定律，稱剛體定軸轉(zhuǎn)動(dòng)定律)Torque(力矩)Torque

for

body

rotating

about

a

Fixed

AxisNewton’s

2nd

Law

for

RotationWork

for

Rotation??~????

theorem

for

RotationUniversity

Physics

I2Motion

of

Translation

vs.motion

of

Rotation(平動(dòng)與轉(zhuǎn)動(dòng))If

an

object

moveswithoutspinning,

the

motion

ispure

translational.If

an

object

spins

aboutan

axis,

the

motion

ispure

rotational.axisUniversity

Physics

I34Motion

of

a

Rigid

Body（剛體的運(yùn)動(dòng)）A

body

is

rigid

if

each

point

of

the

body

has

fixed

relative

distances

with

others.The

sh of

a

rigid

body

is

locked

during

movement.1.Translational

motion(平動(dòng)):For

a

rigid

body

in

puretranslational

motion,everypoint

of

the

body

movesthrough

the

same

lineardistance

in

a

particular

timeinterval.2.Rotational

motion(or

angular

motion,轉(zhuǎn)動(dòng)):In

pure

rotation

about

a

fixed

axis

(called

rotation

axis),every

point

ofthe

body

moves

in

a

circle

whose

centerlies

on

the

axis

of

rotation,

and

every

pointmoves

through

the

same

angle

during

a

particular

timeinterval.University

Physics

I5A

rigid

body(剛體)is

an

large

object,consists

ofmanyparticles,and

is

regarded

as

a

system

ofparticles(質(zhì)點(diǎn)系)In

real

life,

objects

are

large

and

they

behave

like

rigid

bodiesmore

than

like

particles!A

particle(質(zhì)點(diǎn))only

shows

translational(linear)motion.It

does

not

have

rotational

motion(or

angular

motion)A

rigid

body

can

haveboth

translational

motion

and

rotational

motion.

Its

translational

motioncan

be

represented

by

the

motionof

its

COM(center

of

mass)Four

Angular

variables (or

rotational

variables)—4個(gè)有關(guān)“轉(zhuǎn)動(dòng)運(yùn)動(dòng)學(xué)”的變量In

thekinematics

of

rotational

motion(or

angularmotion,轉(zhuǎn)動(dòng)運(yùn)動(dòng)學(xué)),we

also

have

fourequivalent

rotational

variables(or

angular

variables)of

Angular

position,Angulardisplacement,angular

velocity,and

angular

acceleration(角位置,角位移,角速度,角加速度).Fourvariableskinematics

oftranslational

motion(or

linear

motion),1Position

:

??2Displacement:

???3Velocity

:

??4Acceleration

:

??In

thekinematics

of

translationalmotion(or

linear

motion,平動(dòng)運(yùn)動(dòng)學(xué)),we

have

four

majortranslational

variables(or

linear

variables)of

position,displacement,velocity,andacceleration(位置,位移,速度,加速度).-----see

Chapter1-4University

Physics

I7(1).Angular

position

(角度位置??)The

angular

position

θ

of

the

reference

line

is

the

angle

of

the

line

relative

to

afixed

direction

(saying

x

axis),

which

we

take

as

the

zero

angular

position.The

angle

is

positive

if

it

is

counterclockwise(反時(shí)針)with

respect

to

the

positive

x

axis.Units

of

Angular

Position

??The

SI

unit

of

θ

is

radian

(rad), which

is

a

purenumber.??8revolution(圈,轉(zhuǎn))l:

length

of

arc,

??

=

????一圈的弧長（周長??=??????）2??1rad=360°

=57.3°度2??=

1

rev=0.159rev(圈)(2).

Angular

Displacement(???角位移)An

angular

displacement

in

thecounterclockwise

direction

ispositive,and

one

in

the

clockwisedirection

is

negative.University

Physics

I9(3).

Angular

Velocity(??

角速度)The

instantaneousangular

velocity

is

:The

SI

unit

of

??

is

rad/sThe

angular

velocity

??

is

positiveif

the

rotation

is

conterclockwise(4).

Angular

Acceleration(角加速度??)University

Physics

I10Are

Angular

ties

Vectors

?NO!

If

the

rotation

axis

varies.??

+

??

=

??

+

????1

+

??2

≠

??2

+

??1??1??2??1??1

+

??2≠??2

+??1??2University

Physics

I11Right-hand

rule:

We

canuse

the

right-hand

rule

topoint

to

the

directions

ofangular

displacement

???,angular

velocity

??,

andangular

acceleration

??.Are

Angular ties

Vectors

?Yes!

If

the

rotation

axis

is

fixed.12猿人vs人？平動(dòng)vs

轉(zhuǎn)動(dòng)？University

Physics

I13translational(orlinear)motion平動(dòng)rotational

motion(orangularmotion)轉(zhuǎn)動(dòng)????Velocity(線)速度??Acceleration(線)加速度????angularvelocity角速度??angular

acceleration角加速度??tThe

rotational

motion

(or

angular

motion)

compares

to

the

translationalmotion(or

linear

motion),平動(dòng)與轉(zhuǎn)動(dòng)的對(duì)比translational(or

linear)motion平動(dòng)rotational

motion(or

angular

motion)轉(zhuǎn)動(dòng)??

(質(zhì)量)??

(轉(zhuǎn)動(dòng)慣量)??

(加速度)??

(角加速度)??

=

????Linear

momentum(動(dòng)量)??

=

????angular

momentum(角動(dòng)量)??force(力)??Torque(力矩)The

2nd-law:??

=

????The

2nd-law:??

=

????The

2nd-law:??????

=

????(=????)The

2nd-law:??

=

????????(=

????)2.1

Kinematics(運(yùn)動(dòng)學(xué)的比較)

2.

2

Dynamics(動(dòng)力學(xué)的比較)University

Physics

I143.Rotation

with

constant

angular

acceleration(勻角加速轉(zhuǎn)動(dòng))Study

of

rotational

motion

is

easy!It

is

similar

to

linear

motion!

(learned

in

Chapters

2&4,

see

Lecture

2)Translational

motion(along

a

straight

line)with

constant

linear

acceleration）（中學(xué)物理）勻(線)加速直線運(yùn)(平)動(dòng)速度：??=??0

+????2位移：??=??0??+1

????2??2

?

??

2

=

2?? ??

?

??0

0Let

??0

=

0Rotational

motion

(or

angular

motion)

withconstant

angular

acceleration勻角加速轉(zhuǎn)動(dòng)

(大學(xué)物理)??

=

??0

+

????012??

=

??

??

+

????2??2

?

??02

=

2?? ??

?

??0Let

??0

=

015中學(xué)物理勻(線)加速直線運(yùn)(平)動(dòng)大學(xué)物理勻角加速轉(zhuǎn)動(dòng)??

=

??0

+

??????

=

??0

+

??????

?

?? =

??

??

+

1

????

20

0

2??

??? =

??

??

+

1

????

20

0

2??2

?

??

2

=

2?? ??

?

??0

0??2

?

??02

=

2?? ??

?

??0Conclusion:不必頭暈！Study

of

rotational

motion

iseasy!It

is

similarto

the

linear

motion

learned

in

high

school!Chapter

4Chapter

10Derivation:Rotation

with

constant

angular

acceleration

??

(勻角加速轉(zhuǎn)動(dòng))??

=

????????????

=??????0??

?

??????

=????

??????

?

??0

=

????0

012??

?

?? =

?? ??

+

????2??2

?

??02

=

2?? ??

?

??0??

is

constanteliminate

t????

=

????????????????

=????

????

=????

=

??University

Physics

I16Sample

Problem

10-1A

grindstone

rotates

atconstant

angular

acceleration??

=

0.35rad/s2.At

time

??

=

0,

it

has

an

angularvelocity

of

??0

=

?4.6

rad/s

,and

a

reference

line

on

it

ishorizontal,

at

the

angularposition

??0

=

0.??0

=

?4.6

rad/sUniversity

Physics

I17(a)

At

what

time

after

??

=

??

is

the

reference

line

at

theangular

position

??

=

??.

??

rev?Solution:??

=

??.

??

rev=5

×2??

=

10??

radUniversity

Physics

I18(b)

Describe

the

grindstone’s

rotation

between

t

=

0and

t

=

32

s.University

Physics

I19(c)

At

what

time

t

does

the

grindstone

momentarily

stop?Solution:

“momentarily

stop”

means

at

this

moment

“t”??

=

0University

Physics

I20Sample

Problem

10-2While

you

are

operating

a

Rotor

(the

rotatingcylindrical

ride

discussed

in

Sample

Problem6-5

of

chapter

6),

you

spot senger

inacute

distress

and

decrease

the

angularspeed

of

the

cylinder

from

3.40

rad/s

to

2.00rad/s

in

20.0

rev,

at

constant

angularacceleration.General

Physics

I21Solution:(a)

What

is

the

constant

angular

acceleration

duringthis

decrease

in

angular

speed???

=20.0

rev=20

×

2??

=

125.7??????(b)

Howmuch

time

did

the

speed

decrease

take?Solution:University

Physics

I22University

Physics

I233.1

Relating

Linear

and

Angular

tiesThe

position:

the

point

P

moves

the

distanceequaling

to

the

length

of

the

circular

arc

“s”The

speed

alongtangent

??

(切向速率):If

??

varies,

??

moves

in

a

non-uniformcircular

motion.If

??

is

constant,

the

period

of

revolution:??

=

2????University

Physics

I24??????

????

????Tangential

acceleration(切向加速度):??=

????

=

??(????)

=

??

????

=

??????

=

????

+

????Radial

(Centripetal)

acceleration:(徑向加速度,向心加速度)????

???? =

??2

=

(????)2

=

??2??

=

??

??????

= ??2

+

??2??

??Linear

Acceleration

of

a

Point

in

Circular

Motion25LineartyAngulartyL-A

relation??????

=

??????????

=

??????????????

=

??????????2????

=

??

=

??2??Relating

Linear

and

AngulartiesLinearAngular??????????

=??????????

=????????????

=

??????????

=????Once

again(第幾次？):Sample

Problem

5-4Like

ndulum(單擺)，a

small

bob

isreleased

from

horizontal

level

(??

=

0°).??

=

0°????????1.

What

are

the

speed

??,

the

acceleration

??

ofthe

ball

when

the

cord

sweeps

an

angle

of

???????????????The

tangential

acceleration(切向加速度):

????

=??

cos

??,2??????

sin

??

=

1????2??

= 2????

sin

??????The

centripetal

acceleration(向心加速度):

??

=

??2

=

2??

sin

??=

??

cos

??）(or

????

=

??????????

= ??2

+

??2

=

?? 1

+

3

sin2

????

??when

??

=

0°, ??

=

??when

??

=

90°, ??

=

2??26??

=

0°????????????2.

What

are

the

angular

speed

ω,

the

angular

acceleration

??

?????????The

tangential

acceleration(切向加速度):????

=??

cos

??,????The

centripetal

acceleration(向心加速度):

??

=

??2

=

2??

sin

????

2??

sin

????

= 2????

sin

??????

=

=??

=

??

=????????

cos

????????

????27(or

??

=→?

)????(or??

=

????

→?

)ω??Sample

Problem

10-3The

figure

showsa

centrifuge(離心機(jī))used

toaccustomastronaut

trainees

tohighaccelerations.The

radius

ofthe

circle

traveledby

anastronaut

is

??=15

m.At

what

constant

angularspeed

must

the

centrifuge

rotate

if

theastronaut

to

have

a

radial

linearacceleration

of

magnitude

????

???

28General

Physics

I29Solution:Because

the

angular

speed

isconstant,

only

the

radiallinear

acceleration

is

present.??

=

????

=

??2??University

Physics

I30Solution:(b)

What

is

the

tangential

acceleration

of

the

astronautif

the

centrifuge

accelerates onstant

rate

fromrest

tothe

angular

speed

of

(a)

in

120

s?Tangential

acceleration

????is

much

smaller

than

the

radial

acceleration

??.0??

=

?? +

??????

=

??

???0??314.Kinetic

Energy(????)of

Rotation(轉(zhuǎn)動(dòng)動(dòng)能)We

treat

a

rotating

rigid

body

(about

a

fixedrotation

axis)

as

a

collection

ofparticles.Each

particle

has

a

mass

????

and rpendicular

distance

????

from

the

rotationaxis.Each

particle

has

the

same

angular

velocity

??.

Each

particle

has

the

differentlinear

speed

????

=

??????.The

total

kinetic

energy

of

the

rigid

body:4.1

Rotational

Inertia(Moment

of

Inertia)&

????(轉(zhuǎn)動(dòng)慣量與轉(zhuǎn)動(dòng)動(dòng)能)Rewrite:Kinetic

energy

dueto

rotation:WhereI

is

called

“Rotational

inertia”

for

a

rigid

body

about

a

particular

axis:SI

unit

of

the

scalar

??:

kg?

m2A

smaller

??

means

easier

to

rotatethe

body????

=

??????University

Physics

I32Sample

Problem

10-4The

figure

showsa

rigid

bodyconsisting

of

two

particles

of

mass??

connected

by

a

rod

of

length

??and

negligible

mass.(a)

What

is

the

rotational

inertia

????????

of

the

body

about

anaxis

through

its

center

of

mass

and

perpendicular

to

the

rod.Solution:comUniversity

Physics

I33Or,

we

can

use

the

parallel-axis

theorem(will

be

learned

later)(b)

What

is

the

rotational

inertia

of

the

body

about

an

axisthrough

the

left

end

of

the

rod

and

parallel

to

the

axis?Solution:University

Physics

I344.2

Calculating

the

Rotational

Inertial(Moment

of

Inertial)Rotational

inertiais

determined

by

the

massdistribution

of

a

body

relative

to

the

rotation

axis.The

mass

of

a

rod

isdistributedmuch

closer

totherotation

axisin

case

(a).

So

the

rotationalinertia

of

the

rod

is

smaller

incase

(a)

and

the

rod

iseasiertorotate

around

that

axis.University

Physics

I35Rotational

inertia

about

a

given

rotation

axis

for

bodies

with

Continuous

massdistribution:??

=

??2????University

Physics

I36All

cases

in

have

rotationaxes

through

the

center

ofmass

(com)

of

the

bodies,so

gave

the

rotationalinertia

of

????????The

rotational

axes

arethrough

the

“com”,

but

may

along

differentdirections,

e.g.

cases

in

(a)and

(h).Sample

problems:

Rotational

inertia

about

a

given

rotation

axis

for

bodies

withContinuous

mass

distribution:

??

=

??2????(i)

Calculate

the

rotational

inertia

????????

in

cases

(a),(b),(c)

and

comparewith

them.In

all

cases,rotation

axes

are

through

the

center

of

mass

(com)

of

the

body,

so

we

have

I

=

????????In

(a):

????????

=In

(c):

????????

=??2????

= ??2????

=

??2

????

=

??2????2??????=

??2?????? ?

????

=??????2????22?????? ?

????

=

2??

??????2??????

=

??????

=

??

2????

?

????

?

??Therefore

????????

= ??2????

=??2??=????3

????

=??=02????2??4412?

=

????2University

Physics

I37????????Sample

problems:

Rotational

inertia

about

a

given

rotation

axis

for

bodies

withContinuous

mass

distribution:

??

=

??2????In

the

two

cases,

the

objects

are

the

same,

but

they

rotate

about

different

rotation

axesthrough

the

center

of

mass

(com)

of

the

body,

and

have

different

rotational

inertia

????????In

(a):

????????

= ??2????

= ??2????

=

??2

????

=

??2??In

(h):

????????

= ??2????,

??

=

??

cos

??????????

=????=0Calculate

the

rotational

inertia

????????

in

cases

(a),(h)

and

compare

with

them.y??

=

??

cos??

????

=

??????????

=

??????

=

??

????????=

2??????=2????2????

= (??

cos

??)2?

2??

????

=????????=

????2??????22????=2??

1

?

cos

2??2??=0????

=????2

12??

22University

Physics

I382??

=

1????2 4.3

Parallel-Axis

Theorem(平行軸定理)

comThe

rotational

inertia

I

of

a

body

varies

as

therotational

axis

changes.??comis

the

rotational

inertia

about

the

axis

through

its

center

of

mass

(com).????

=

????????

+

???2namely,

Parallel-axis

theoremA

body

with

mass

??

has

a

rotational

inertia??com

about

a

axis

through

its

center

of

mass.Given

a

rotation

axis,

which

is

parallel

to

theaxis

through

the

center

of

mass

inaperpendicular

distance

?,

the

rotationalinertia

???of

the

body

about

this

given

axis

is:???University

Physics

I39??comUniversity

Physics

I40Proof

of

Parallel-Axis

TheoremSet

the

origin

at

the

center

of

mass.??=

??com?2

=

??2

+

??2,???

=

??2??????Now

the

rotation

axis

is

through

“??(??,

??)”??= (??2

+

??2

+

??2

+

??2

?

2????

?

2????)????= ??2

+

??2

????

+ ??2

+

??2

????

?

2?? ??????

?

2b??????+?2????com

=??2????

= ??2

+

??2????2

+= ??

??? ??

?

??

2

????????

=

????????

+

???2Then??com

=??????=0,

??com

=??????=0????????

=

??Sample

problems:

Parallel-Axis

Theorem,

Rotational

inertia

about

different

rotation

axes

for

the

same

body.

(i)

Calculate

the

rotational

inertia

????????

in

cases

(a)

and

(b),

and

compare

with

them.(b)In

the

two

cases,

the

objects

are

the

same,

but

they

rotate

about

different

rotationaxes

parallel

to

each

other,

so

they

have

different

rotational

inertia

??In

(a):

????????

= ??2????

= ??2????

=

??2

????

=

??2??In

(b):

??

= ??2????

=

????????

+

???2

=

2????2University

Physics

I415.1

Torque

(

力矩？)：5. Newton’s

2nd

Law

for

Rotation

about

a

Fixed

Axis(剛體定軸轉(zhuǎn)動(dòng)的

第二定律，or

called

剛體定軸轉(zhuǎn)動(dòng)定律)Torque

??

on

a

particle

(body),

whichis

rotating

about

a

fixed

point(origin

??),

due

to

a

force

??

appliedon

it

at

a

position

??

is

defined

as:??

=

??

×

??Torque

is

avector ty

withSI

unit

of

N

?

m,the

same

as

work.

But

work

is

a

scalar.Direction

of

torque

follows

right-hand

rule,

??

isperpendicular

to

????????

??Magnitude

of

torque

??

=

????

sin

??.Net

torque

??net

by

several

forces

is

the

sum

ofindividual

torques.力矩=力X力臂University

Physics

I425.2

Torque

for

a

body

rotating

about

a

Fixed

Axis1.

Any

force

acting

on

a

body

rotating

about

afixed

axis

can

be posed

into

twocomponents:

one

is

??,

being

perpendicular

torotation

axis

(i.e.

in

the

plane

perpendicular

to

the

rotation

axis),

and

the

other

component

is

??‖

,being

parallel

to

the

rotation

axis.??‖

is

not

under

our

consideration,

because

it

doesnot

affect

therotation,

butonly

causes

the

rotationaxis

tilting

away.??‖沿任意方向force????????????2.

We

only

need

to

consider

??

(lying

on

the

plane

perpendicularto

the

rotation

axis),

which

affects

the

body

rotating

about

the

axis.??

is

thenposed

into

????

(radial

component

along

??)

and

????(tangential

component

perpendicular

to

??):??

=

????

+

????43??⊥University

Physics

I44??

is

zero

if

extended

line

of

force

passes

through

axis.Direction:

Torque(along

the

same

direction

of

angular

velocity

??)

ispositive

(pointing

up),

if

the

rotation

is

counterclockwise.

It

is

negative

ifdriven

rotation

is

clockwise.總外力矩455.3

Newton’s

2nd

Law

for

RotationParticle

“i”

with

mass

????

is

part

of

a

body,

experiences

a

net

force

????o????

=

????????,??

=

????????????The

rotating

speed

of

each

particle

????

=

???????? ??

??2?? =

(??

??

)??????????=

??

??????

=

??

=????

???? ??

=

??????net,??????

=

????Newton’s

2nd

law

forrotation

about

a

fixed

axis剛體定軸轉(zhuǎn)動(dòng)定律Internal

forces

between

immediate

neighboring

particles

form

Newton’s

3rd

force

pair.Torques

due

to

paired

internal

forces

cancels

each

other.

??net,internal

=

0.Net

torque

takes

only

from

external

applied

forces

on

the

body.??=

??

??????????????質(zhì)量與轉(zhuǎn)動(dòng)慣量的對(duì)比Mass

compares

to

rotational

inertiaM

（質(zhì)量）??平動(dòng)慣量轉(zhuǎn)動(dòng)慣量??

=

??????

=

??????

=

????Linear

momentum(動(dòng)量)??

=

????angular

momentum(角動(dòng)量)質(zhì)量大：讓物體由“靜到轉(zhuǎn)動(dòng)慣量大：讓物體轉(zhuǎn)動(dòng)時(shí)由“靜到動(dòng)”(或“由動(dòng)到靜”)需要的力大！轉(zhuǎn)”(或“由轉(zhuǎn)到靜”)需要的力矩大！質(zhì)量?。鹤屛矬w由“靜到動(dòng)”(或“由動(dòng)到靜”)需要的力小！轉(zhuǎn)動(dòng)慣量?。鹤屛矬w轉(zhuǎn)動(dòng)時(shí)由“靜到轉(zhuǎn)”(或“由轉(zhuǎn)到靜”)需要的力矩??！(

：質(zhì)量與速度有關(guān))轉(zhuǎn)動(dòng)慣量??

=

??????????與轉(zhuǎn)動(dòng)物體的質(zhì)量及所取的轉(zhuǎn)軸有關(guān)46University

Physics

ISample

Problem

10-5Find

the

magnitude

of

the

tensions

????,

????

on

the

two

blocks,

themagnitude

of

the

acceleration

??

of

the

blocks,

in

terms

of

??,

??,

??,

??.??2??1????Revisit

the

sample

problem

5-3

of

chapter

5.(where

the

pulley

is

massless).onBut,

here

now

the

pulley

has

a

radius

??

anda

rotational

inertia

??

about

the

frictionlesshorizontal

axel

due

to

its

mass.Released

from

the

rest,

the

hanging

block

Hfalls

as

the

sliding

block

S

accelerates

to

theright

without

the

taut

massless

cord

slipthe

pulley.??

=?47University

Physics

Ig

13

NMmT

g

3

.

8

m

/

s

2M

mmM

ma

Previously

in

Chapter

5,

the

pulley

is

massless!48University

Physics

I????ponent

of

block

S:T

Maponent

of

block

H:????

?

??

=

????Two

unknown

variables

，T

and

??，can

be

solvedSolution:???? ?

??1

=

????

(2)??2

=

????

(1)Now

look

at

the

pulley:

Force

from

axle

and

the

gravitational

force

pass

throughrotation

axis,

give

zero

torque.????????????????2??1??????????

=

??1??

?

??2??

=

????

(3)Newton’s

2nd

law

for

the

rotation

of the

pulley????For

block

S????2

??????Newton’s

2nd

law

on

two

blocks:For

block

H??1??The

pulley

rotates

clockwise

due

to

the

torque

from

tensions

??1

and

??2

.??2??1

??????

49University

Physics

INonslip

cord:

the

acceleration

of

the

cord,

with

thesame

magnitude

of

blocks’

acceleration,

is

equal

tothe

tangential

acceleration

????

of

the

pulleyrim??

=

????

=

????

(4)Eq.

(1)-(4)

give:??

=????

+??

+??/??2??,??

+??/??2??1

=

??

+

??

+

??/??2

????,????2

=

??

+

??

+

??/??2

????Bigger

??,

smaller

??.??2

=

????

(1)????

?

??1

=

????

(2)??1??

?

??2??

=

????

(3)????????????50University

Physics

I共三個(gè)未知數(shù):??1，??2，??(??=??/??)Discussion:The

same

results

as

the

massless

pulley

case

of

sample

problem

in

chapter

5.??1

=

??

+

??

+

??/??2

????2??

=????

+

??

+??/??2??????

=????

+??

+??/??2????1

=

??2=

????????

+??If

the

pulley

is

massless,????

=

0,

so

that

??

=

0,??

+

??/??2????

+

????

=

??51University

Physics

ISample

Problem

10-6Block

1

of

mass

??1

and

block

2

of

heavier

mass

??2(??2

>

??1)

are

connected

by

a

massless

cordaround

a

pulley

of

radius

??,

which

is

mounted

on

ahorizontal

axle

with

a

rotational

inertia

??

andnegligible

friction.When

released

from

rest,

the

block

2

falls

and

the

block

1rises

without

the

taut

cord

slip on

the

pulley.Considering

at

any

moment

after

releasing

and

beforeany

block

hits

the

pulley,Find

the

magnitude

of

the

tensions

????,

????

on

the

two

blocks,

themagnitude

of

the

acceleration

??

of

the

blocks,

the

magnitude

of

thepulley’s

angular

acceleration??,

in

terms

of

????,

????,

??,

??.52University

Physics

I53Solution:Connected

by

a

taut

cord,

two

blocks

have

the

samemagnitude

of

acceleration

and

the

same

speed.Newton’s

2nd

law

on

two

blocks:??1

?

??1??

=

??1??

(1)??1??1??????2??2??????2??

?

??2

=

??2??

(2)????Newton’s

2nd

law

on

the

pulley:

Force

from

axle

and

the

gravitationalforce

pass

through

rotation

axis,

give

zero

torque.????????????2??1??????????

=

??2??

?

??1??

=

????

（3）??1????2??University

Physics

IThe

pulley

rotates

clockwise

due

to

the

torque

fromtwo

tensions.Nonslip

cord:

the

magnitude

of

acceleration

ofthe

cord

and

the

blocks

is

equal

to

the

magnitudeof

tangential

acceleration

????

of

pulley

rim.??

=

????

=

????

(4)2??2

+

??/??2??1

=

??1

+

??2

+

??/??2

??1??2??1

+

??/??2??2

=

??1

+

??2

+

??/??2

??2????

=??2

?

??1??1

+??2

+

??/??2??,??

=

??2

?

??1

????1

+

??2

+

??/??2

??Bigger

??,smaller

??

and

??.??????????University

Physics

I54??共三個(gè)未知數(shù):??1，??2，??(??=??/??)Eq.

(1)-(4)

give:Discussion:The

same

results

as

the

massless

pulley.If

the

pulley

is

massless,

??

=

0,2??2

+

??/??2??1

=

??1

+

??2

+

??/??2

??1??2??1

+

??/??212??2

=

??+?? +

??/??

2

??2??1?? =

??2=

2??1??2????1

+??2??

=??2

?

??1??1

+

??2

+??/??2????1

+

??2??

=

??2

?

??1

??University

Physics

I55General

Physics

I56Sample

Problem

10-7Find

the

acceleration

of

the

falling

block,

the

angularacceleration

of

the

disk,

and

the

tension

in

the

cord.The

figure

shows

a

uniform

disk,

with

mass

M

=

2.5

kg

andradius

R

=

20

cm,

mounted

on

a

fixed

horizontal

axle.Rotational

inertial

of

the

disk

about

the

axis

is

??

=

1????2.2A

block

withmass

m

=1.2

kg

hangs

from

a

massless

cordthat

is

wrapped

around

the

rim

of

the

disk.Rel