Chapter 9.Chapter 9.ppCenter of mass and linear momentumCenter of mass and linear momentum

질량중심과질량중심과 선운동량선운동량질량중심과질량중심과 선운동량선운동량

질량중심 (Center of Mass)

선운동량 (Linear momentum)

충돌 (C lli i )충돌 (Collision)

충격량 (Impulse)충격량 (Impulse)

Physics, Page 1

QuestionIs it possible for a system of two objects to have

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

1. YES

2 NO

correct

2. NO

f l h ff f h h h 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.

Physics, Page 2

Question You drop an egg onto

1) the floor 2) a thick piece of foam rubber1) the floor 2) a thick piece of foam rubber.

In both cases, the egg does not bounce.

In which case is the impulse greater?In which case is the impulse greater?

A) Floor

B) FJ = ΔP

B) Foam

C) the sameSame change in momentum

In which case is the average force greater

A) FloorA) Floor

B) Foam

C) th

J = Δp = F ΔtF = Δp/Δt

Physics, Page 3

C) the same

24

Smaller Δt = large F

Physics, Page 4

1i imrr mr= =∑ ∑

rr rcm i i

i

r mrm M

= = ∑∑y

mr m r+r r

xCM1 1 2 2

1 2cm

mr m rrm m

+=

+r

m1m2

x+

xCM

x1x2

Physics, Page 5

rmrmrmr ii∑∑ =⋅⋅⋅++

=rrr

r 2211

MmmrCM ∑ =

⋅⋅⋅++=

21

∑ ∑ zm∑M

xmx iiCM

∑= Mymy ii

CM∑=

Mzmz ii

CM∑=

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Physics, Page 7

Fnet = 질량이질량이 M M 인인 단일입자처럼단일입자처럼 취급취급

: p mv=ur r

(kg·m/sec.)

d pF F∑ur

ur ur( )d p d dm dv dv⎧ ⎫⎛ ⎞⎪ ⎪⎛ ⎞

⎜ ⎟⎨ ⎬⎜ ⎟

ur r rr r r

net i

pF Fdt

= =∑ ( )p mv v m m madt dt dt dt dt

⎪ ⎪⎛ ⎞= = + = =⎜ ⎟⎨ ⎬⎜ ⎟⎝ ⎠⎪ ⎪⎝ ⎠⎩ ⎭

P + + + +ur ur ur ur ur

: CM P M v=ur r

1 2 3

1 2 31 2 3

n

n CMn

P p p p p

m v m v m v m v M v

= + + + +

= + + + + =

Lr r r r r

L

netd PFdt

=ur

ur CMCM

dvM M adt

⎧ ⎫⎪ ⎪= =⎨ ⎬⎪ ⎪⎩ ⎭

ur rrd P

dt

Physics, Page 8

dt dt⎪ ⎪⎩ ⎭dt

No net force ⇒ Momentum is conserved.

∑ ==pdaMF totr

rr∑ ==

dtaMF CMext

∑ 0Fr

0=ar이면

CM ith t t l it

∑ = 0extF 0=CMa이면

CM moves with a constant velocity.

Physics, Page 9

9-6 충돌과 충격량

충돌의 정도를충격량 (impulse)으로 표기

Physics, Page 10

충격량 (Impulse)( p )

어느 정도의 힘을얼마 시간 동안 받았나얼마 시간 동안 받았나?

Physics, Page 11

충격량충격량 JJ

Physics, Page 12

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 fl I 1 th b ll b b k d i 2 th b llfloor. 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

correct

C. The same Bouncing Ball Sticky Ball

|J| = |Δp|

= |mv m v |

|J| = |Δp|

= |mv m v |= |mvf – m vi|

= |m( vf - vi)|

= |mvf – m vi|

= |m ( 0 - vi)|

Physics, Page 13

= 2 m v = m v

QuestionQuestion In both cases of the above question the direction ofIn 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

correct

time

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Physics, Page 15

(Conservation of Linear Momentum)( )

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For a Two-particle system

111 vmp rr=

2112 FFrr

−= : Newton’s third law

0∑ FFFrrr

222 vmp rr=1m

12Fr

21Fr

02112 =+=∑ FFF

21 pppptotrrrr

+== ∑

2m

21

0211221 =+=+= FF

dtpd

dtpd

dtpd tot

rrrrr

1 2totp p p = Δ Δ Δ= +r r r

0

1 2totp p p = = +r r r

일정

ffii pppp 2121rrrr

+=+∴

Physics, Page 17

⇒ Conservation of Total Linear Momentum

Physics, Page 18

9-9 일차원 비탄성 충돌 (Inelastic collision)

Physics, Page 19

보기문제 9-8: 탄동진자

(즉, 충돌 후의 운동과정에서는 역학에너지 보존)

Physics, Page 20

9-10 일차원 탄성 충돌 (Elastic collision)

Physics, Page 21

v1f > 0, v2f > 0

< 0 > 0v1f < 0, v2f > 0

Physics, Page 22

보기문제 9-10: 탄성진자

3. 4. 도 유사하게.

Physics, Page 23

9-11 이차원 충돌

1 2 1 2i i f fP P P P+ = +ur ur ur ur

1 2 1 2i i f fK K K K+ = +

(예) 비낌 탄성충돌 (정면충돌이 아닌 경우)

Physics, Page 24

문제

정지해 있는 당구공을 향해 질량이 같은 당구공을 탄성 충돌 시켰더니 그림과 같정지해 있는 당구공을 향해 질량이 같은 당구공을 탄성 충돌 시켰더니 그림과 같

이 비껴 나갔다. 충돌 후 둘 사이의 진행 각은 항상 서로 수직임을 증명.

ppf ppi

vvCM

FF PPf

before after

CM

1 1 2 2 )i f f 1v v v ( m m= + =r r r

Q

2 2 21 1 1i f fv v v= +

제곱

2 2 21 21 1 1 2 f fi f fv v v v v= + +r r

r r

Physics, Page 25

1 2 0f fv v∴ ⋅ =r r

에 … 운동량과 운동에너지가 모두 보존되는 탄성충돌이니 … 서로 90도를 이루겠구먼 …

그러니까 … 힌 공을 A로 가도록 하면 빨간 공은 B로 가고힌 공을 A로 가도록 하면 빨간 공은 B로 가고 …

A

B

Physics, Page 26

로켓 연료

(dM < 0)연료

Physics, Page 27

Rocket Propulsion

Momentum conservation

Lift-off of the space shuttle.

( ) ( ) ( )evvmvvMvmM rrrrr−Δ+Δ+=Δ+

emvvM Δ=Δ e

dMMd

dMdmm −=→Δ

dMvMdv e−=

∫∫ −= ff M

Mev

v MdMvdv ∫∫

ii Mev M

⎟⎟⎠

⎞⎜⎜⎝

⎛=− i

eif MMlnvvv ⎟

⎠⎜⎝ fM

Thrust =d

dMvddvM e=

Physics, Page 28

(추진력) dtdt e

Thrust > Mg for acceleration

SSSummarySummaryI l J FΔ– Impulse J = FΔt• Gives change in momentum J = Δp

– Momentum p = mv• Momentum is VECTOR• Momentum is conserved

Σ mv = Σ mv (when ΣF = 0)Σ mvi Σ mvf (when ΣF 0)– Center of Mass

+ rr

∑+=

icm m

rmrmr 2211rr

r

Physics, Page 2950