¾ 회전 운동에너지

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Physics, Page 1

Chapter 10. Chapter 10. Rotation ( Rotation ( 회전 회전 ) )

¾ 회전운동 변수들

z각도(θ), 각변위(Δθ), 각속도(ω) , 각가속도(α)

¾ 회전 운동에너지

¾ 회전관성 (관성모멘트, moment of inertia)

¾ 돌림힘 (Torque)

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Review of last lecture Review of last lecture

F_aveΔt ≡ J = p_f - p_i = Δp

¾ 충격량 - 선운동량 정리 (Impulse-Momentum Theorem)

¾ 단일 입자인 경우 …

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

¾ 여러 입자 계인 경우 …

tf ( )

ti

P_total ≡ Σp_i

Internal forces: forces between objects in system External forces: all other forces

F_extΔt = ΔP_total

if F_ext = 0 , then total momentum conserved (ΔP_total = 0)

¾ 탄성충돌 (운동량, 운동에너지 보존) 비 탄성충돌 (운동량만 보존) J ≡

∫

F t dt = Δp

ur ur ur

F d p

= dt ur ur

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Physics, Page 3

변수비교

병진운동 (linear motion)

회전운동

(angular motion)

변위 (displacement) x m θ

ω α

I (1/2)Iω²

τ = Iα

- (rad)

속도 (velocity) v

m(r x v)

rad /s 가속도 (acceleration) a

m/s m/s²

kg J N

rad/s²

관성 (inertia) m kg^.m²

운동에너지 (KE) (1/2)mv² J

운동방정식 (Newton’s 2^nd) F = ma N^.m

N^.s

모멘텀 (momentum) mv J^.s

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Physics, Page 5

(부호 약속 : 반시계 방향 +, 시계방향 -)

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Comment on axes and sign of angle Comment 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 usually called positive.

Clockwise (decreasing θ) is usually called negative.

z

+ω

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Physics, Page 7

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(선변수와 각변수의 관계)

r s a_t

v a_r

즉즉, , 가속도의가속도의관점만관점만 다르다다르다..

a_r = rω² ÅÆ a_t = αr

* ^지름방향

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Physics, Page 9

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1 2 2

2 2

1 2

1 2 2

2 ( )

( )

o

o o

o

t

t t

t t t

ω ω α

θ θ ω α ω ω α θ θ θ θ ω ω θ θ ω

= +

= + +

= + −

= + +

= + −

( , , ; )

angular t θ ω α

1 2 2

2 2

1 2

1 2 2

2 ( )

( )

o

o o

o

v v at

x x v t at

v v a x x

x x v v t

x x vt t

= +

= + +

= + −

= + +

= + −

( , , ; )

linear x v a t

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Physics, Page 11

(= 관성모멘트)

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보기문제 : Oxygen Molecule

d

( )

¹₂ ²

( )

₂¹ ² ⁴⁶ ²

2 m d m d 1.95 10 kg m

r m

I = ∑_i _i _i = _i + _i = × ⁻ ⋅

d = 1.21 × 10^-10m m_i= 2.66 × 10^-26 kg

ω = 4.6 × 10¹²rad/sec.

J I

K = ₂¹

ω

² = 2.06×10⁻²¹

x

회전운동에너지 회전운동에너지

? ?

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Physics, Page 13

Inertia rods

Two batons have equal mass and length.

Which will be “easier” to spin?

A) Mass on ends B) Same

C) Mass in center

I = Σ m r²Further mass is from axis of rotation, greater moment of inertia (harder to spin)

21

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회전관성 회전관성 ( ( 관성모멘트 관성모멘트 ) ) 계산법 계산법

{ } { }

2

2 2

2 2 2 2

2

( ) ( )

( ) 2 2 ( )

( ) ( )

P

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

=

⎡ ⎤

= ⎣ − + − ⎦

= + − − + +

= + + +

= +

∫

∫ ∫ ∫ ∫

∫ ∫

dm (x,y)

CM (0,0)

P (a,b)

P

h

P

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Physics, Page 15

보기 : Cylinder 의 회전관성 ^R

l

r

( )

l R M M

l

R

²

₂

ρ π

π

ρ ⋅ ⋅ = ⇒ ∴ = Density ρ

( )

{ ^r ^dr ^l } ^r ^lr ^dr

r dm

dI = ⋅

²

= ρ ⋅ 2 π ⋅ ⋅ ⋅

²

= 2 πρ

³

(

²

)

² ²

4 2

1 0

3

2 1 2

2 l r dr lR 1 R l R MR

I = ^πρ ∫

^R

= ^ρπ = ^ρπ =

증명 2

1 ₂

MR I_CM =

M : mass

Z Z’

I

_z’

= ?

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몇 가지 물체의 회전관성

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Physics, Page 17

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.

24

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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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Physics, Page 19

돌림힘 (회전력, Torque)

( ^sin )

r F r F r F

τ r = × r ur = φ =

_⊥

r : _⊥ 모멘트 팔(의 길이)

( ^sin )

t

r F r F rF

τ r = × r ur = φ =

F : t 힘의 접선성분

F_t

r

r_ㅗ F

τ r r ur = × r F

[ ^{N m} ]

_{= ⎣}^⎡

^{kg m}

² ^/

^s

²^⎤_⎦

r F

] / [

]

[ N ⋅ m = kg ⋅ m

²

s

²

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☞ Vector Product

θ

sin AB C

C = = Cr Ar Br , ⊥

B A

C

In (x,y,z)-coordinates iˆ×iˆ = jˆ× jˆ = kˆ×kˆ = 0

jˆ kˆ iˆ iˆ

kˆ

iˆ jˆ kˆ kˆ

jˆ

kˆ iˆ jˆ jˆ

iˆ

=

×

−

=

×

=

×

−

=

×

=

×

−

=

×

r r r = ×

Scalar Product and Vector Product

kˆ b jˆ b iˆ b B

kˆ a jˆ a iˆ a A

3 2

1

3 2

1

+ +

=

+ +

r =

r Scalar Product ≡ Ar⋅Br = A B cosθ

(

â1îˆ + â2 ^jˆ + â3^kˆ

) (

⋅ ^b1^iˆ +^b2 ^jˆ +^b3^kˆ

)

=

3 3 2

2 1

1b a b a b

a + +

=

Vector Product ≡ Ar×Br = A B sinθcˆ ₍ ₎ B , A cˆ r r

(

â1îˆ +â2 ^jˆ +â3^kˆ

) (

× ^b1^iˆ +^b⊥2 ^jˆ +^b3^kˆ

)

=

(

^a ^b ⁻^a ^b

) (

^kˆ ⁺ ^a ^b ⁻^a ^b

) (

^jˆ ⁺ ^a ^b ⁻^a ^b

)

^iˆ

=

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Physics, Page 21

τ τ = r = × r F r ur

( )

( ) ( )

²

sin

t t

r F rF r ma rm r

mr I

τ φ

α

α α

=

= =

net

I τ = α

r s a_t

v a_r

a

_t

= α r

(a_r = rω² ^{와는 다르다}⁾

F_r = ma_r 와는 다르다

⋅

제2법칙

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τω

=

= dt P dW

⇒ Power

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Physics, Page 23

보기문제 10-8 : 추의 가속도는?

τ =

물체의 가속도 (a) 방향을

각가속도 (α) 방향 (+θ 인 반시계방향) 과 동일하게 잡았다.

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h

R M

vrCM

ω

Kinetic Energy of Rolling object

2 2

1 ω

= I_P K

Kinetic Energy of the Disc.

MR2

I

I_P = _CM +

U K

K U

K

E_T = _T + = _R_,_CM + _L_,_CM + Parallel axis theorem

2 2 2

2 1 2

1 ω + ω

= I MR

K _CM

2 2

2 1 2

1

CM

CM Mv

I

K = ω +

CM을 중심으로

하는 질량중심의

선형운동에너지

2 2

2 1 2 1

CM

T Mgh I Mv

E = = ω +

Total Energy

보기문제

vrCM

ω

P• R

매 순간을 보면 P 점을 중심으로 회전하고 있다.

¾ 회전 운동에너지

Chapter 10.

Chapter 10. Rotation ( Rotation ( 회전 회전 ) )

¾ 회전운동 변수들

z각도(θ), 각변위(Δθ), 각속도(ω) , 각가속도(α)

¾ 회전 운동에너지

¾ 회전관성 (관성모멘트, moment of inertia)

¾ 돌림힘 (Torque)

Review of last lecture Review of last lecture

∫

변수비교

Comment on axes and sign of angle Comment on axes and sign of angle

(i.e. what is positive and negative) (i.e. what is positive and negative)

t

t t

t t t

ω ω α

θ θ ω α ω ω α θ θ θ θ ω ω θ θ ω

angular t θ ω α

v v at

x x v t at

v v a x x

x x v v t

x x vt t

linear x v a t

d

( )

( )

ω

? ?

Inertia rods

Two batons have equal mass and length.

Which will be “easier” to spin?

A) Mass on ends B) Same

C) Mass in center

회전관성 회전관성 ( ( 관성모멘트 관성모멘트 ) ) 계산법 계산법

{ } { }

∫

∫

∫ ∫ ∫ ∫

∫ ∫

( )

l R M M

l

R

ρ π

π

ρ ⋅ ⋅ = ⇒ ∴ = Density ρ

( )

{ r dr l } r lr dr

r dm

dI = ⋅

= ρ ⋅ 2 π ⋅ ⋅ ⋅

= 2 πρ

(

)

2 1 2

2 l r dr lR 1 R l R MR

I = πρ ∫

= ρπ = ρπ =

I

= ?

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

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

돌림힘 (회전력, Torque)

( sin )

r F r F r F

τ r = × r ur = φ =

( sin )

r F r F rF

τ r = × r ur = φ =

τ r r ur = × r F

[ N m ]

kg m

{ ^r ^dr ^l } ^r ^lr ^dr

I = ^πρ ∫

= ^ρπ = ^ρπ =

( ^sin )

( ^sin )

[ ^{N m} ]

^{kg m}

^s