Physics, Page 1

Chapter 10.Chapter 10. Rotation (Rotation (회전회전))

회전운동 변수들

각도(θ), 각변위(Δθ), 각속도(ω) , 각가속도(α)

회전 운동에너지

회전관성 (관성모멘트, moment of inertia)

돌림힘 (Torque)

Physics, Page 2

Review of last lectureReview of last lecture

FaveΔt ≡ J = pf - pi = Δp

충격량 - 선운동량 정리 (Impulse-Momentum Theorem)

단일 입자인 경우 …

If F = 0, then momentum conserved (Δp = 0)

여러 입자 계인 경우 …

Ptotal ≡ Σpi

Internal forces: forces between objects in system

External forces: all other forces

FextΔt = ΔPtotal

if Fext = 0 , then total momentum conserved (ΔPtotal = 0)

탄성충돌 (운동량, 운동에너지 보존) 비 탄성충돌 (운동량만 보존)

( )tf

tiJ F t dt pΔ≡ =∫

d pFdt

=

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QuestionYou and a friend are playing on the merry-go-round at Carle Park. You stand at the outer edge of the merry-go-round and your friend stands halfway between the outer edge and the center. Assume the rotation rate of the merry-go-round is constant.

Who has the greatest angular velocity?

1. You do2. Your friend does 3. Same CORRECT

Because within the same amount of time you and your friend both travel 2π.

Objects that are father away from the axis move faster. The larger the radius, the smaller the angular velocity. ω = v/r

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Who has the greatest tangential velocity?

1. You do 2. Your friend does 3. Same

CORRECT

In one rotation, the person on the outside is covering more distance in the same amount of time as the one on the inside. This means it's a faster speed.

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Question

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변수비교

m(r x v)

τ = Iα

(1/2)Iω2

I

α

ω

θ

회전운동

(angular motion)

N.s

N

J

kg

m/s2

m/s

m

N.mF = ma운동방정식 (Newton’s 2nd)

J.smv모멘텀 (momentum)

J(1/2)mv2운동에너지 (KE)

kg.m2m관성 (inertia)

1/s2a가속도 (acceleration)

1/sv속도 (velocity)

-x변위 (displacement)

병진운동

(linear motion)

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(부호 약속 : 반시계 방향 +, 시계방향 -)

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Comment on axes and sign of angleComment on axes and sign of angle(i.e. what is positive and negative)(i.e. what is positive and negative)

Whenever we talk about rotation, it is implied that there is a rotation “axis”.

This is usually called the “z” axis (we usually omit the z subscript for simplicity).

Counter-clockwise (increasing θ) is usuallycalled positive.

Clockwise (decreasing θ) is usuallycalled negative.

z

+ω

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212

2 2

12

212

2 ( )( )

o

o o

o o

o o

o

t

t t

t

t t

ω ω α

θ θ ω α

ω ω α θ θθ θ ω ω

θ θ ω

= +

= + +

= + −

= + +

= + −

( , , ; )angular tθ ω α

212

2 2

12

212

2 ( )( )

o

o o

o o

o o

o

v v at

x x v t at

v v a x xx x v v t

x x vt t

= +

= + +

= + −

= + +

= + −

( , , ; )linear x v a t

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(선변수와 각변수의 관계)

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(= 관성모멘트)

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보기문제 : Oxygen Moleculed

( ) ( ) 2462212

212 10951 mkg.dmdmrmI iii ii ⋅×=+== −∑

d = 1.21 × 10-10 mmi = 2.66 × 10-26 kgω = 4.6 × 1012 rad/sec.

JIK 21221 1006.2 −×== ω

x

회전운동에너지회전운동에너지??

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Inertia rodsTwo batons have equal mass and length. Which will be “easier” to spin?

A) Mass on endsB) SameC) Mass in center

I = Σ m r2 Further mass is from axis of rotation, greater moment of inertia (harder to spin)

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회전관성회전관성 ((관성모멘트관성모멘트) ) 계산법계산법

{ } { }

2

2 2

2 2 2 2

2 2 2 2

2

( ) ( )

( ) 2 2 ( )

( ) ( )

CM

I r dm

x a y b dm

x y dm a xdm b ydm a b dm

x y dm a b dm

I Mh

=

⎡ ⎤= − + −⎣ ⎦

= + − − + +

= + + +

= +

∫∫∫ ∫ ∫ ∫∫ ∫

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몇 가지 물체의

회전관성

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보기 : Cylinder 의 회전관성R

l

r

( )lR

MMlR 22

πρπρ =∴⇒=⋅⋅

Density ρ

( ){ } drlrrldrrrdmdI 322 22 πρπρ =⋅⋅⋅⋅=⋅=

( ) 222421

0

3

21

212 MRRlRlRdrrlI

R==== ∫ ρπρππρ

증명 21 2MRICM =

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Rolling Race (Hoop vs Cylinder)

A hoop and a cylinder of equal mass roll down a ramp with height h. Which has greatest KE at bottom?

A) Hoop B) Same C) Cylinder

“they both start with the same potential energy so they have to end with the same kinetic energy because of conservation of energy.

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Rolling Race (Hoop vs Cylinder)

A hoop and a cylinder of equal mass roll down a ramp with height h. Which has greatest speed at the bottom of the ramp?

A) Hoop B) Same C) Cylinder

Cylinder will get to the bottom first because inertia for a cylinder is less than that for a hoop type object.

I = MR2 I = ½ MR2

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돌림힘 (회전력, Torque)

r Fτ = ×

( )sin tr F r F rFτ φ= × = =

( )sinr F r F r Fτ φ ⊥= × = =

r : ⊥ 모멘트팔(의 길이)

tF : 힘의접선성분

[ ] 2 2/N m kg m s⎡ ⎤= ⎣ ⎦i i

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☞ Vector Product

θsinABCC == BAC , ⊥

In (x,y,z)-coordinates 0=×=×=× kkjjii

jkiik

ijkkj

kijji

=×−=×

=×−=×

=×−=×

BAC ×=

Scalar Product and Vector Product

kbjbibB

kajaiaA

321

321

++=

++= Scalar Product θ=⋅≡ cosBABA( ) ( )kbjbibkajaia 321321 ++⋅++=

332211 bababa ++=

Vector Product csinBABA θ=×≡ ( )B,Ac⊥

( ) ( )kbjbibkajaia 321321 ++×++=

( ) ( ) ( )ibabajbabakbaba 233231131221 −+−+−=

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r Fτ τ= = ×

( )

( )( )

( )2

sin

t

t

r FrFr ma

rm r

mr I

τ φ

α

α α

=

=

=

=

= =

net Iτ α=

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보기문제 10-8 : 추의 가속도는?

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τω==dt

dWP⇒ Power

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h

RM

CMv

ω

Kinetic Energy of Rolling object

221 ω= PIK

Kinetic Energy of the Disc.

2MRII CMP +=

UKKUKE CM,LCM,RTT ++=+=

Parallel axis theorem

22212

21 ω+ω= MRIK CM

2212

21

CMCM MvIK +ω=KR about CM KL

2212

21

CMT MvIMghE +ω==

Total Energy

CMvω

P•

R

보기문제