9. Systems of Particles?????

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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 ????

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As the skier flies through the air,
??????????, most parts of his body follow
complex trajectories. ?????????????? But one
special point follows a parabola. ????????????????
? Whats that point, and why is it
special? ??????,??????
Ans. His center of mass (CM) ? ????
Rigid body Relative particle positions fixed.
?? ?????????
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9.1. Center of Mass ??
N particles ??
?
total mass ???
Center of mass ?? mass-weighted average
position ?????????
with
3rd law ? ????
Cartesian coordinates ????
Extension particle i may stand for an
extended object with cm at ri . ?? ??
i ????? CM ? ri ??????
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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. ???????1.5 m
,?????????????? 50-kg ? 80-kg ??? ???????
CM is closer to the heavier mass. ????????????
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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. ?????????,??????,???
L ???,????????? 2 modules have mass m, the other
2m. ??????? m,???? 2m ? Find the CM. ????
Coord origin at m2 2m y points
downward. ????? m2 2m ? y ???
2 2m
x
30?
L
CM
obtainable by symmetry ???????
1 m
3m
y
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Continuous Distributions of Matter ???????
Discrete collection ????
Continuous distribution ????
????
Let ? be the density of the matter. ? ? ??????
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Example 9.3. Aircraft Wing ???
A supersonic aircraft wing is an isosceles
triangle of length L, width w, and negligible
thickness. ?????????????????,? L ,? w ,??????? It
has mass M, distributed uniformly. ???? m
????? Wheres its CM? ???????
Density of wing ???? ?.
Coord origin at leftmost tip of wing. ??????????
By symmetry ????,
y
dx
h
w
x
L
8
y
b
dy
w/2
w
x
?w/2
L
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CMfuselage ??
CMplane ??
CMwing ??
A high jumper clears the bar, but his CM
doesnt. ??????????,????????
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Got it ??? 9.1.
A thick wire is bent into a semicircle. ??????????
? Which of the points is the CM? ???????
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Example 9.4. Circus Train ?????
Jumbo, a 4.8-t elephant, is standing near one end
of a 15-t railcar, ?? 4.8-t ???,??,???? 15-t
???????? which is at rest on a frictionless
horizontal track. ???????????????? Jumbo walks 19
m toward the other end of the car. ?????????? 19m
? How far does the car move? ???????
1 t 1 tonne ?? 1000 kg
Car not moving
Car moved
xcm ?
19xJi
?
Jumbo walks, but the center of mass doesnt move
(Fext 0 ). ?????,?????? ( Fext 0 ) ?
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9.2. Momentum ??
Total momentum ???
M constant ? ??
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Conservation of Momentum ????
?
Conservation of Momentum ???? Total momentum
of a system is a constant if there is no net
external force. ???????,?????????????
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GOT IT ??! 9.2.
A 500-g fireworks rocket is moving with velocity
v 60 j m/s at the instant it explodes. ?? 500-g
??????????? v 60 j m/s ?????? If you were to
add the momentum vectors of all its fragments
just after the explosion, ???????????????????????
??, what would you get? ??????
K.E. is not conserved. ??????? Emech K.E.
P.E. grav is not conserved ????. Etot
Emech Uchem is conserved ??.
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Example 9.5. Kayaking ????
Jess (mass 53 kg) Nick (mass 72 kg) sit in a
26-kg kayak at rest on frictionless water. ??
(??53 kg) ??? (?? 72 kg) ????????????, 26-kg
?????? Jess toss a 17-kg pack, giving it a
horizontal speed of 3.1 m/s relative to the
water. ??????17-kg ???,??????? 3.1 m/s
?????? Whats the kayaks speed while the pack is
in the air after Nick catches
it? ??????,?????????,?????????
Initially ???
While pack is in air ??????
After Nick catches it ???????
Note Emech not conserved ? Emech ???
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Example 9.6. Radioactive Decay ?????
A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s
when it decays into ???-5 ( 5Li ) ???? 1.6 Mm/s
?????? a proton ( 1H, or p ) an alpha particle
( 4He, or ? ). Superscripts denote mass in
AMU ????( 1H, ? p ) ????????( 4He,? ? ).
????? AMU?????? ? is detected moving at 1.4
Mm/s at 33? to the original velocity of 5Li. ???
? ? 1.4 Mm/s ?? 5Li ????? 33? ?????? What are the
magnitude direction of ps velocity? p
????????????
Before decay ????
After decay ????
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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. ???????? 45 kg/s ,????????32 m/s ????? After
hitting the window, the water drops
vertically. ?????,???????? What horizontal force
does the water exert on the window? ??????????????
?
Momentum transfer to a plane ? stream ? ?
??????,?????
Rate of momentum transfer to window
??????????
force exerted by water on window ???????
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GOT IT ?? ? 9.3.

• Two skaters toss a basketball back forth on
  frictionless ice.
• ???????????????????????
• Which of the following does not change
• ?????????
• momentum of individual skater. ????????
• momentum of basketball. ?????
• momentum of the system consisting of one skater
  the basketball.
• ????????????????
• momentum of the system consisting of both
  skaters the basketball.
• ????????????????

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Application Rockets ????
High pressure gas pushes equally in opposite
directions, this rocket goes
nowhere. ???????????????,??????????
Open an exhaust port . ???????
and theres now an unbalanced force on the front
of the rocket. ???????????????
Thrust ??
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9.3. Kinetic Energy of a System ???????
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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 )
• ???????????

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Momentum in Collisions ??????
External forces negligible ? Total momentum
conserved ??????? ? ?????
For an individual particle ????
?t collision time ????
impulse ??
More accurately,??????
Same size ????
Average ???
Crash test ????
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Energy in Collisions ??????
Elastic collision K conserved. ???? K ??
? Inelastic collision K not conserved. ????? K
??? ?
Bouncing ball inelastic collision between ball
ground. ??????????????
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GOT IT ??? 9.4.

• Which of the following qualifies as a
  collision ??????????
• Of the collisions, which are nearly elastic
  which inelastic?
• ????,?????????,????????
• a basketball rebounds off the backboard.
• ???????????
• two magnets approach, their north poles facing
  they repel reverse direction without touching.
  ???????????,??????,??????????
• a basket ball flies through the air on a
  parabolic trajectory.
• ????????????????
• a truck crushed a parked car the two slide off
  together.
• ??????????????,?????????
• a snowball splats against a tree, leaving a lump
  of snow adhering to the bark.
• ?????????,???????????

elastic??
elastic ??
inelastic???
inelastic???
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9.5. Totally Inelastic Collisions ???????

• Totally inelastic collision colliding objects
  stick together
• ?????????????????
• maximum energy loss consistent with momentum
  conservation.
• ???????????????

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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. ???????????????,???????????? 6.4 kg? A 160-g
puck strikes gets embedded in the chest, which
moves off at 1.2 m/s. ??160-g ?????,????,????
1.2 m/s ??? What is the pucks speed? ????????
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Example 9.9. Fusion ??
Consider a fusion reaction of 2 deuterium nuclei
2H 2H ? 4He . ???????????? 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. ???,?? 2H ? 3.5 Mm/s ??,????? 1.8 Mm/s
????? 64? ??? Find the velocity of the Helium
nucleus. ????????
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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. ???? m ???????????? M
???? The block swings upward to a vertical
distance of h. ??????? h? Find the bullets
speed. ???????
?
Caution??
( heat is generated when bullet strikes block )
???????????
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9.6. Elastic Collisions ????
Momentum conservation ????
Energy conservation ????
Implicit assumption particles have no
interaction when they are in the initial or final
states. ( Ei Ki )
?????????????????? ( Ei Ki )
2-D case ???? number of unknowns ????? 2 ? 2
4 ( final state ?? v1fx , v1fy , v2fx ,
v2fy ) number of equations ????? 2 1 3 ?
1 more conditions needed. ????????
3-D case ???? number of unknowns ????? 3 ? 2
6 ( final state ?? v1fx , v1fy , v1fz ,
v2fx , v2fy , v2fz ) number of equations ?????
3 1 4 ? 2 more conditions needed. ????????
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Elastic Collisions in 1-D ??????
1-D case ???? number of unknowns ????? 1 ? 2
2 ( v1f , v2f ) number of equations ?????
1 1 2 ? unique solution. ???????
1-D collision ????
This is a 2-D collision ??????
?
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?
?
(a) m1 ltlt m2
?
(b) m1 m2
?
(c) m1 gtgt m2
?
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Example 9.11. Nuclear Engineering ????
Moderator slows neutrons to induce
fission. ???????????????? A common moderator is
heavy water ( D2O ). ?????????( D2O ) ? Find the
fraction of a neutrons kinetic energy thats
transferred to an initially stationary D in a
head-on elastic collision. ????????,??????????????
?????? D ?
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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. ??????
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Elastic Collision in 2-D ??????
Impact parameter ???? b additional info
necessary to fix the collision outcome. ??????????
?????
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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. ??????,??????????????? 30?
? In what direction does the other ball
move? ??????????
p cons ??
E cons ??
?
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Center of Mass Frame ???
???