질량중심과

Chapter 9.

Chapter 9.

Center of mass and linear momentum Center of mass and linear momentum

질량중심과

질량중심과 선운동량 선운동량

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Is it possible for a system of two objects to have zero total momentum while having a non-zero total kinetic energy?

1. YES 2. NO

If two people sitting on separate carts at rest push off of each other the total momentum will be zero because they go in opposite directions. But the kinetic energy will not be zero since they are now moving.

correct

Question

You drop an egg onto

1) the floor 2) a thick piece of foam rubber.

In both cases, the egg does not bounce.

In which case is the impulse greater?

A) Floor B) Foam C) the same

In which case is the average force greater A) Floor

B) Foam C) the same

J = Δ P

Same change in momentum

J = Δ p = F Δ t F = Δ p/ Δ t

Smaller Δ t = large F

Question

물체물체 : : 입자계입자계(system of particles), (system of particles), 크기크기(size)(size)

i i

1

cm i i

i

r m r m r

m M

= ∑ ∑ = ∑

r r r

1 1 2 2

1 2

cm

m r m r

r m m

= +

+

r r

r

m₁ m₂

x₁

x x₂

+ x_CM

y

M r m m

m

r m r

r ^r

⁼ ∑ m ^{r +} ₊ ^r ₊ ⁺ _⋅ ^⋅ _⋅⋅ ^{⋅⋅ =} ∑

^r

2 1

2 2 1

1

M x x

₌ ∑ m

M y y

₌ ∑ m

M

z

z

₌ ∑ m

F

=

입자계에 입자계에 대한 대한 뉴턴의 뉴턴의 제법칙 제법칙

: p ur = mv r

1 2 3

1 2 3

1 2 3

n

n CM

n

P p p p p

m v m v m v m v M v

= + + + +

= + + + + =

ur ur ur ur ur

r r L r r r

L

: P ur = M v r

net i

F F d p

= ∑ = dt ^ur

ur ur ^{⎧ ⎪ ⎨ d p} _dt ⁼ _dt ^d ( ) ^{mv = ⎛ ⎜} _⎝ ^dm _dt ^{⎞ ⎟} _⎠ ^{v + m ⎛ ⎜ d v} _dt ^{⎞ ⎟ = m d v} _dt ^{= ma ⎫ ⎪ ⎬}

⎪ ⎝ ⎠ ⎪

⎩ ⎭

ur r r

r r r

(kg·m/sec.)

net

F d P

= dt

ur ur

CM

M d v M a dt

⎧ ⎫

⎪ = = ⎪

⎨ ⎬

⎪ ⎪

⎩ ⎭

ur r r

d P dt

No net force ⇒ Momentum is conserved.

∑ ^{= =}

dt p a d

M

F

r r

r

CM moves with a constant velocity.

∑ F r

^{= 0}

= 0 a r

이면

9-6 충돌과 충격량

충돌의 정도를 충격량 (impulse) 으로 표기

충격량 (Impulse)

어느 정도의 힘을

얼마 시간 동안 받았나?

충격량충격량 JJ

Question

Two identical balls are dropped from the same height onto the floor.

In each case they have velocity v downward just before hitting the floor. In case 1 the ball bounces back up, and in case 2 the ball sticks to the floor without bouncing. In which case is the magnitude of the impulse given to the ball by the floor the biggest?

A. Case 1 B. Case 2 C. The same

correct

Bouncing Ball

|J| = | Δ p|

= |mv

– m v

|

= |m( v

- v

)|

= 2 m v

Sticky Ball

|J| = | Δ ^p|

= |mv

– m v

|

= |m ( 0 - v

)|

= m v

Question

In both cases of the above question, the direction of the impulse given to the ball by the floor is the same.

What is this direction?

•A. Upward

•B. Downward

time correct

(Conservation of Linear Momentum)

~ For a Two-particle system

1 1

1 m v

pr = r

2 2

2 m v

pr = r m1

m2

Fr12

Fr21 21

12 F

Fr r

−

= : Newton’s third law

21 0

12 + =

∑

2

1

p

p p

p r

= ∑ r = r + r

21

0

12 2

1

+ = + =

= F F

dt p d dt

p d dt

p

d r

r r r r

1 2

p r

= p r + p r =

⇒ Conservation of Total Linear Momentum f

f i

i p p p

p r ₁ + r ₂ = r ₁ + r ₂

∴

1 2

p

p p =

Δ r = Δ r + Δ r 0

9-9 일차원 비탄성 충돌 (Inelastic collision)

보기문제 9-8: 탄동진자

(즉, 충돌 후의 운동과정에서는 역학에너지 보존)

9-10 일차원 탄성 충돌 (Elastic collision)

v_1f < 0, v_2f > 0 v_1f < 0, v_2f > 0

보기문제 9-10: 탄성진자

3. 4. 도 유사하게.

9-11 이차원 충돌

1i 2i 1f 2f

P + P = P + P ur ur ur ur

1i 2i 1f 2f

K + K = K + K

(예) 비낌 탄성충돌 (정면충돌이 아닌 경우)

문제

정지해 있는 당구공을 향해 질량이 같은 당구공을 탄성 충돌 시켰더니 그림과 같 이 비껴 나갔다. 충돌 후 둘 사이의 진행 각은 항상 서로 수직임을 증명.

1ⁱ 1^f 2 ^f ₁ 2

)

v r = v r + v r ( m = m Q

2 2 2

1i 1f 1f

v = v + v

2 2 2

1 2

1i 1f 1f

2

v = v + v + v r r v

1f 2 f

0

v v

∴ r ⋅ r =

pp_f pp_i

FF PP_f

before after

vv_CM

제곱

에 …

운동량과 운동에너지가 모두 보존되는 탄성충돌이니 … 서로 90도를 이루겠구먼 …

그러니까 …

힌 공을 A로 가도록 하면 빨간 공은 B로 가고 … A

B

(dM < 0)

로켓 연료

Lift-off of the space shuttle.

Rocket Propulsion

( ^{M + Δ m} ) ^{v r = M} ( ^{v r + Δ v r} ) ^{+ Δ m} ( ^{v r − v r}

)

Momentum conservation

mv

v M Δ = Δ

dM v

Mdv = −

dM dm

m → = − Δ

∫

∫

v M

v dM dv

⎟⎟

⎠

⎞

⎜⎜

⎝

= ⎛

−

f i e

i

f M

ln M v v

v

Thrust =

(추진력) dt

v dM dt

M dv = _e

Thrust > Mg for acceleration

Summary Summary

– Impulse J = F Δ t

• Gives change in momentum J = Δ p

– Momentum p = mv

• Momentum is VECTOR

• Momentum is conserved

Σ mv

= Σ mv

(when ΣF = 0)

– Center of Mass

∑

= +

i

cm

m

r m r

r m

r

r r