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

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 …

P_total ≡ Σp_i

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

F_extΔt = ΔP_total

if F = 0 , then total momentum conserved (ΔP = 0)

tf ( )

J ≡

∫

F d p

= dt

Question

You 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 do

2. 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

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.

Question

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^.m F = ma

운동방정식 (Newton’s 2^nd)

J^.s 모멘텀 (momentum) mv

J (1/2)mv²

운동에너지 (KE)

kg^.m² 관성 (inertia) m

1/s² 가속도 (acceleration) a

1/s 속도 (velocity) v

- 변위 (displacement) x

병진운동 (linear motion)

(부호 약속 : 반시계 방향 +, 시계방향 -)

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

1 2 2

2 2

1 2

1 2 2

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 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 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

(선변수와 각변수의 관계)

(= 관성모멘트)

보기문제 : Oxygen Molecule

d

( )

( )

2 m d m d 1.95 10 kg m

r m

I =

∑

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

ω = 4.6 × 10¹²rad/sec.

J I

K = ₂¹

ω

x

회전운동에너지

회전운동에너지 ? ?

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 r

Further mass is from axis of rotation,

greater moment of inertia (harder to spin)

회전관성 회전관성 ( ( 관성모멘트 관성모멘트 ) ) 계산법 계산법

{ } { }

2

2 2

2 2 2 2

( ) ( )

( ) 2 2 ( )

I r dm

x a y b dm

x y dm a xdm b ydm a b dm

=

⎡ ⎤

= ⎣ − + − ⎦

= + − − + +

∫

∫

∫ ∫ ∫ ∫

몇 가지 물체의

회전관성

보기 : Cylinder 의 회전관성

l

r

( )

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

( ) ¹

1

R

= = =

= ^πρ ∫ ^{ρπ ρπ}

증명 2

1

MR I_CM

=

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.

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

I = MR² I = ½ MR²

돌림힘 (회전력, Torque)

τ = × r F

( ^sin )

r F r F rF

τ = × = φ =

( ^sin )

r F r F r F

τ = × = φ =

r : _⊥

모멘트 팔(의 길이)

힘의 접선성분

[ ^{N m i} ] _{= ⎣} ^{⎡ kg m i}

^{/ s}

^⎤ _⎦

☞ Vector Product

θ

C = = 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

=

×

−

=

×

=

×

−

=

×

=

×

−

=

×

B A

C = ×

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

+ +

=

+ +

= Scalar Product ≡ A⋅B = A B cosθ

(

) (

)

=

3 3 2

2 1

1b a b a b

a + +

=

Vector Product ≡ A×B = A B sinθcˆ ⁽ cˆ⊥A,B ⁾

( ) ( )

τ τ = = × r F

( )

( )

( ) ( )

sin

t t

r F rF r ma rm r

mr I

τ φ

α

α α

=

=

=

=

= =

net

I

τ = α

보기문제 10-8 : 추의 가속도는?

τω

=

= dt P dW

⇒ Power

h

R M

vCM

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

Total Energy

vCM

ω

P• R

보기문제