9. Systems of Particles

1. Center of Mass

2. Momentum

3. Kinetic Energy of a System

4. Collisions

5. Totally Inelastic Collisions

6. Elastic Collisions

As the skier flies through the air,

most parts of his body follow complex

trajectories.

But one special point follows a parabola.

What’s that point, and why is it special?

Rigid body: Relative particle positions fixed.

Ans. His center of mass (CM)

9.1. Center of Mass

i i imF a1

N

total ii

F F2

21

Ni

ii

dm

dt

r 2

21

N

i ii

dm

dt

r

2

2

i imdMdt M

r

1

1 N

cm i ii

mM

r r2

2

i icm

md

dt M

ra

2

2cmd

dt

r

1

Next int

total i ii

F F F1

Nexti

i

F1

0N

inti

i

F3rd law netF

1

1 N

cm i ii

x m xM

1

1 N

cm i ii

y m yM

1

1 N

cm i ii

z m zM

Cartesian coordinates:

1

N

ii

M m

= total mass

rcm = Center of mass = mass-weighted average position of the collection of particles.

net cmMF a

Extension: “particle” i may stand for an extended object with cm at ri .

Example 9.1. Weightlifting

Find the CM of the barbell consisting of 50-kg & 80-kg weights

at opposite ends of a 1.5 m long bar of negligible weight.

1 1 2 2

1 2cm

m x m xx

m m

2 2

1 2

m x

m m

80 1.5

50 80

kg m

kg kg

0.92 m

CM is closer to the heavier mass.

Example 9.2. Space Station

A space station consists of 3 modules arranged in an equilateral triangle,

connected by struts of length L & negligible mass.

2 modules have mass m, the other 2m.

Find the CM.

Coord origin at m2 = 2m & y points downward.

1 1 10

4 2 2cmx L

2 4M m m m m

1 1

1, , cos30

2x y L L

1 3,

2 2L

2 2, 0 , 0x y

1 3 30

4 2 2cmy L

3

4L 0.43L

0

1

1 N

cm i ii

mM

r r

obtainable by symmetry

2: 2m

1: m 3:m

L

x

y

CM

30

2 2

1, , cos30

2x y L L

1 3,

2 2L

Continuous Distributions of Matter

1

1 N

cm i ii

mM

r r

01

1limi

N

cm i im

i

mM

r r

1

N

ii

M m

Continuous distribution:

Discrete collection:

01

limi

N

im

i

M m

dm

1dm

M r

Let be the density of the matter.

M dV r 1cm dV

M r r r dm dV r

Example 9.3. Aircraft Wing

A supersonic aircraft wing is an isosceles triangle of length L, width w, and negligible thickness.

It has mass M, distributed uniformly.

Where’s its CM?

Density of wing = .

dm h dx / 22w

x dxL

0

LwM x dx

L

1

2w L

2

0

L

cm

wx x dx

M L

By symmetry, 0cmy

31

3

wL

M L 22

3L

L

Wx

y

dx

Coord origin at leftmost tip of wing.

wx dxL

/ 2 2

L wdm y dy

w

/2

0

22

2

wL wM y dy

w

2 2

12 2

2 2 2

L w w

w

1

2w L

/2

/2

20

2

w

cm w

L wy y y dy

w

L

Wx

y

dy

w/2

w/2

A high jumper clears the bar,

but his CM doesn’t.

CMfuselage

CMwing

CMplane

Got it? 9.1.

A thick wire is bent into a semicircle.

Which of the points is the CM?

Example 9.4. Circus Train

Jumbo, a 4.8-t elephant, is standing near one end of a 15-t railcar,

which is at rest on a frictionless horizontal track.

Jumbo walks 19 m toward the other end of the car.

How far does the car move?

1 t = 1 tonne = 1000 kg

Jumbo walks, but the center of mass doesn’t move (Fext = 0 ).

J J i c cicm i

m x m xx

M

J cM m m

J J f c c fcm f

m x m xx

M

19J c f ci Ji c c fm x x x m m x

M

cm i cm fx x

4.6 m 19J

c f ciJ c

m mx x

m m

9.2. Momentum

Total momentum:i

i

P p ii

i

dmdt

ri i

i

dm

dt

r cm

dM

dt r

cm

dMd

t

rPM constant cmM v

cmd

dt

dM

dt

vPcmM a net extF

Conservation of Momentum

net ext

d

dt

PF

0net ext F constP

Conservation of Momentum:

Total momentum of a system is a constant if there is no net external force.

GOT IT! 9.2.

A 500-g fireworks rocket is moving with velocity v = 60 j m/s at the instant it explodes.

If you were to add the momentum vectors of all its fragments just after the explosion,

what would you get?

ˆ ˆ0.5 60 / 30 /kg m s kg m sj j

Example 9.5. Kayaking

Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water.

Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water.

What’s the kayak’s speed while the pack is in the air & after Nick catches it?

Initially 0 0p

While pack is in air:

1 1 0( ) 0J N k p pp m m m v m v p

1p

pJ N k

mv v

m m m

173.1 /

55 72 26

kgm s

kg kg kg

0.35 /m s

After Nick catches it:

2 0v

2 2 1( ) 0J N k pp m m m m v p

Example 9.6. Radioactive Decay

A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s when it decays into

a proton ( 1H, or p ) & an alpha particle ( 4He, or ). [ Superscripts denote mass in AMU ]

is detected moving at 1.4 Mm/s at 33 to the original velocity of 5Li.

What are the magnitude & direction of p’s velocity?

Before decay:0 Li LimP v , 0Li Lim v

After decay: 1 p pm m P v v

1 cos , sinp p x p p ym v m v m v m v P

cosLi Li p p xm v m v m v 0 sinp p ym v m v

1cosp x Li Li

p

v m v m vm 1

5 1.6 / 4 1.4 / cos331.0

u Mm s u Mm su

3.3 /Mm s

sinp yp

mv v

m

4 1.4 / cos33

1.0

u Mm s

u

3.05 /Mm s

2 2p p x p yv v v 4.5 /Mm s

1tan p yp

p x

v

v 43

Example 9.7. Fighting a Fire

A firefighter directs a stream of water to break the window of a burning building.

The hose delivers water at a rate of 45 kg/s, hitting the window horizontally at 32 m/s.

After hitting the window, the water drops horizontally.

What horizontal force does the water exert on the window?

Water loses horizontal momentum completely after hitting window:

45 / 32 /xdPkg s m s

dt 1400 N

= force exerted by window on water

= ( force exerted by water on window )

( water moves in +x direction )

GOT IT? 9.3.

Two skaters toss a basketball back & forth on frictionless ice.

Which of the following does not change:

(a) momentum of individual skater.

(b) momentum of basketball.

(c) momentum of the system consisting of one skater & the basketball.

(d) momentum of the system consisting of both skaters & the basketball.

9.3. Kinetic Energy of a System

ii

K K 21

2 i ii

m v 1

2 i cm i rel cm i reli

m v v v v

2 21 1

2 2i cm i cm i rel i i reli i i

m m m v v v v

2 21

2

1

2 cm i i reli

mM v vi

i

M m

0i cm i rel cm i i reli i

m m v v v v

intcmKK K

21

2 mcm cK M v

2int

1

2 i i reli

K m v

9.4. Collisions

Examples of collision:

• Balls on pool table.

• tennis rackets against balls.

• bat against baseball.

• asteroid against planet.

• particles in accelerators.

• galaxies

• spacecraft against planet

( gravity slingshot )

Characteristics of collision:

• Duration: brief.

• Effect: intense

(all other external forces

negligible )

Momentum in Collisions

External forces negligible Total momentum conserved

For an individual particle t p F t = collision time

J impulse

More accurately, t dt J p F

Energy in Collisions

Elastic collision: K conserved.

Inelastic collision: K not conserved.

Bouncing ball: inelastic collision between ball & ground.

GOT IT? 9.4.

Which of the following qualifies as a collision?

Of the collisions, which are nearly elastic & which inelastic?

(a) a basketball rebounds off the backboard.

(b) two magnets approach, their north poles facing; they repel & reverse

direction without touching.

(c) a basket ball flies through the air on a parabolic trajectory.

(d) a truck crushed a parked car & the two slide off together.

(e) a snowball splats against a tree, leaving a lump of snow adhering to the bark.

elastic

elastic

inelastic

inelastic

9.5. Totally Inelastic Collisions

Totally inelastic collision: colliding objects stick together

maximum energy loss consistent with momentum conservation.

1 1 2 2initial m m P v v 1 2final fm m P v

Example 9.8. Hockey

A Styrofoam chest at rest on frictionless ice is loaded with sand to give it a mass of 6.4 kg.

A 160-g puck strikes & gets embedded in the chest, which moves off at 1.2 m/s.

What is the puck’s speed?

initial p pmP v final p c cm m P v

p cp c

p

m mv v

m

0.16 6.41.2 /

0.16

kg kgm s

kg

49 /m s

Example 9.9. Fusion

Consider a fusion reaction of 2 deuterium nuclei 2H + 2H 4He .

Initially, one of the 2H is moving at 3.5 Mm/s, the other at 1.8 Mm/s at a 64 angle to the 1st.

Find the velocity of the Helium nucleus.

1 2D

fHe

m

m v v v

1 2init Dm P v v final He fm P v

23.5 , 0 1.8 cos64 , sin 64 /

4Mm s

2.14 , 0.809 /Mm s

2 22.14 0.809 /fv Mm s

2.3 /Mm s

1 0.809tan

2.14 21

Example 9.10. Ballistic Pendulum

The ballistic pendulum measures the speeds of fast-moving objects.

A bullet of mass m strikes a block of mass M and embeds itself in the latter.

The block swings upward to a vertical distance of h.

Find the bullet’s speed.

m M

m

v V

21

2embE m M V finalE m M g h

init mP v emb m M P V

2 2V g h 2m M

v g hm

Caution: 21

2init finalE m E v (heat is generated when bullet strikes block)

9.6. Elastic Collisions

Momentum conservation: 1 1 2 2init i im m P v v 1 1 2 2final f fm m P v v

Energy conservation: 2 21 1 2 2

1 1

2 2init i iE m m v v 2 21 1 2 2

1 1

2 2final f fE m m v v

2-D case:

number of unknowns = 2 2 = 4 ( v1fx , v1fy , v2fx , v2fy )

number of equations = 2 +1 = 3

1 more conditions needed.

3-D case:

number of unknowns = 3 2 = 6 ( v1fx , v1fy , v1fz , v2fx , v2fy , v2fz )

number of equations = 3 +1 = 4

2 more conditions needed.

Elastic Collisions in 1-D

1 1 2 2init i ip m v m v 1 1 2 2final f fp m v m v

2 21 1 2 2

1 1

2 2init i iE m v m v 2 21 1 2 2

1 1

2 2final f fE m v m v

1-D case:

number of unknowns = 1 2 = 2 ( v1f , v1f )

number of equations = 1 +1 = 2

unique solution.

1 1 1 2 2 2f i f im v v m v v

2 2 2 21 1 1 2 2 2f i f im v v m v v

1 1 2 2f i f iv v v v

1 2 1 2i i f fv v v v i fv v

2-D collision

1-D collision

1 1 1 2 2 2f i f im v v m v v

1 1 2 2f i f iv v v v

1 2 1 2 21

1 2

2i if

m m v m vv

m m

1 1 2 2 1 1 2 2f f i im v m v m v m v

1 2 1 2f f i iv v v v

1 1 2 1 2

21 2

2 i if

m v m m vv

m m

(a) m1 << m2 : 1 1 22f i iv v v 2 2f iv v

2 0iv 1 1f iv v 2 0fv

(b) m1 = m2 : 1 2f iv v 2 1f iv v

2 0iv 1 0fv 2 1f iv v

(c) m1 >> m2 : 1 1f iv v 2 1 22f i iv v v

2 0iv 1 1f iv v 2 12f iv v

Example 9.11. Nuclear Engineering

Moderator slows neutrons to induce fission.

A common moderator is heavy water ( D2O ).

Find the fraction of a neutron’s kinetic energy that’s transferred to an initially

stationary D in a head-on elastic collision.

1 2 1 2 21

1 2

2i if

m m v m vv

m m

1 1 2 1 2

21 2

2 i if

m v m m vv

m m

1

2

1

2

m u

m u

1 1

2 0

i i

i

v v

v

1

1

1 2

1 2i

f

u u vv

u u

1

1

3 iv

12

2

1 2i

f

u vv

u u

1

2

3 iv

21 1 1

1

2i iK m v

22 2 2

1

2f fK m v

22 2 2

21 1 1

f f

i i

K m v

K m v

2

1

21

22

31

i

i

u v

u v

8

9 89%

GOT IT? 9.5.

One ball is at rest on a level floor.

Another ball collides elastically with it & they move off in the same direction separately.

What can you conclude about the masses of the balls?

1st one is lighter.

Elastic Collision in 2-D

Impact parameter b :

additional info necessary to fix the collision outcome.

Example 9.12. Croquet

A croquet ball strikes a stationary one of equal mass.

The collision is elastic & the incident ball goes off 30 to its original direction.

In what direction does the other ball move?

1 1 2i f f v v vp cons:

2 2 21 1 2i f fv v v E cons:

2 2 21 1 1 2 22i f f f f v v v v v

2 2 21 1 1 2 22 cos 30i f f f fv v v v v

1 22 cos 30 0f fv v

9030 60

Center of Mass Frame

0i f P P