投影片 1 - National Cheng Kung University
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Transcript 投影片 1 - National Cheng Kung University
9. Systems of Particles
1.
2.
3.
4.
5.
6.
Center of Mass
Momentum
Kinetic Energy of a System
Collisions
Totally Inelastic Collisions
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?
Ans. His center of mass (CM)
Rigid body: Relative particle positions fixed.
9.1. Center of Mass
N particles:
Fi mi ai
d 2 ri
d2 N
d 2 rcm
Fi mi
2 m i ri M
2
2
d t i 1
dt
dt
i 1
i 1
N
Ftotal
N
N
M mi
1
rcm
M
= total mass
i 1
N
m r
i 1
i i
= Center of mass
= mass-weighted average position
Ftotal M acm
Ftotal F
N
i 1
ext
i
F
int
i
with
N
F
i 1
ext
i
acm
d 2 rcm
dt 2
Fnet
3rd
law
N
F
i 1
int
i
0
Fnet M acm
Cartesian
coordinates:
1
xcm
M
N
m x
i 1
i
i
1
ycm
M
N
m y
i 1
i
i
1
zcm
M
N
m z
i 1
Extension: “particle” i may stand for an extended object with cm at ri .
i i
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.
xcm
m1 x1 m2 x2
m1 m2
80 kg 1.5 m
50 kg 80 kg
0.92 m
CM is closer to the heavier mass.
m2 x2
m1 m2
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.
1
rcm
M
2 modules have mass m, the other 2m.
Find the CM.
Coord origin at m2 = 2m & y points downward.
x
m r
i 1
i i
1
3
1
L , L cos 30 L ,
2
2
2
x1 , y1
x2 , y2 0
2: 2m
N
, 0
1
L1 , 3
x
,
y
L
,
L
cos
30
3 3
2
2
2
30
L
M m 2 m m 4m
CM
1: m
3:m
y
1 1
1
xcm 0 L 0
4 2
2
obtainable
by symmetry
1 3
3
3
ycm
0
L 0.43L
L
4 2
2
4
Continuous Distributions of Matter
Discrete collection:
1
rcm
M
N
N
m r
i 1
M mi
i i
i 1
Continuous distribution:
M lim
m i 0
N
m
i 1
1
rcm lim
m i 0 M
N
i
dm
mi ri
i 1
1
r dm
M
Let be the density of the matter.
dm r dV
M r dV
rcm
1
r r dV
M
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 = .
Coord origin at leftmost tip of wing.
y
dx
ycm 0
d m h dx x
h
W
L
By symmetry,
x
w
M
L
xcm
L
0
w
ML
x dx
L
0
w
dx
L
h w
x L
1
wL
2
x 2 dx
2
w 1 3
L L
3
M L 3
d m b dy
b
L w
y
dy
w/ 2 2
y
ycm
b
dy
w/2
x
L
w /2 w
2L
M
2 y dy
0
w
2
L
w/ 2
2 L w/2 w
y
y
dy 0
w
/2
w
2
w/2
2L 0
w
w
y
y
dy
y
y
dy
w
/2
0
w
2
2
W
w/2
w
y
2
0
w /2
2 L w 2 1 3
w
1
2
3
y y
y y
w 4
3 w/2 4
3 0
2 L w3 w3 w3 w3
w 16 24 16 24
w 2 1 w 2
L
2
2
w
2 2 2
0
1
wL
2
CMfuselage
CMplane
CMwing
A high jumper clears the bar,
but his CM doesn’t.
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,
1 t = 1 tonne
= 1000 kg
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?
xcm i
mJ xJ i mc xci
xcm f
M
M mJ mc
mJ xJ f mc xc f
M
Final distance of Jumbo from xc : 19 m xci
xJi
xJ f xc f 19 m xci xJi
xcm i xcm f
Jumbo walks, but the center of mass
doesn’t move (Fext = 0 ).
mJ 19m
xc f xci
4.6 m
mJ mc
9.2. Momentum
Total momentum:
P pi mi
i
M constant
i
PM
d ri
d
d
M rcm
mi ri
dt
dt i
dt
d rcm
M vcm
dt
d v cm
dP
M
dt
dt
M acm
Fnet ext
Conservation of Momentum
dP
Fnet ext
dt
Fnet ext 0
P const
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?
0.5 kg 60 ˆj m / s 30 ˆj
kg m / s
K.E. is not conserved.
Emech = K.E. + P.E. grav is not conserved.
Etot = Emech + Uchem is conserved.
Conceptual Example 9.1. 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 after Nick catches it?
Why can you answer without doing any calculations ?
Initially, total p = 0.
frictionless water p conserved
After Nick catches it , total p = 0.
Kayak speed = 0
Simple application of the conservation law.
Making the Connection
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 ?
Initially
p0 0
While pack is in air:
p1 (mJ mN mk )v1 mpvp p0 0
v1
mp
mJ mN mk
vp
17 kg
3.1 m / s 0.35 m / s
55 kg 72 kg 26 kg
Note: Emech not conserved
Example 9.5. 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?
P0 mLi v Li mLi vLi , 0
Before decay:
After decay:
P1 mp v p m v
P1 m p v p x m v cos , m p v p y m v sin
mLi vLi mp vp x m v cos
vp x
1
mLi vLi m v cos
mp
vp y
0 mp vp y m v sin
1
5 u 1.6 Mm / s 4 u 1.4 Mm / s cos33
1.0 u
m
4 u 1.4 Mm / s cos33
3.05 Mm / s
v sin
mp
1.0 u
vp
v v
2
px
2
py
4.5 Mm / s
p tan 1
vp y
vp x
43
3.3 Mm / s
Example 9.6. 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?
Momentum transfer to a plane stream:
dP dm
v 45 kg / s 32 m / s 1400 N
dt
dt
= Rate of momentum transfer to window
= force exerted by water on window
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.
Application: Rockets
Ptot Procket Pfuel const
Thrust:
F v exhaust
dM
dt
9.3. Kinetic Energy of a System
K Ki
i
i
i
1
1
mi vi2 mi v cm vi rel v cm vi rel
2
2
i
1
1
2
mi v cm
mi v cm vi rel mi vi2rel
2
2
i
i
1
1
2
M v cm
mi v i2rel
2
2
i
M mi
i
m
i
K Kcm Kint
K cm
1
M v c2m
2
Kint
i
1
mi vi2rel
2
vcm vi rel vcm mi vi rel
i
i
v cm mi
i
d
ri rcm
dt
v cm
d
mi ri rcm
dt i
vcm
d
M rcm M rcm 0
dt
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
p F t
J
More accurately,
t = collision time
impulse
J p F t dt
Same size
Average
Crash
test
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?
elastic (a) a basketball rebounds off the backboard.
elastic (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.
inelastic
(d) a truck crushed a parked car & the two slide off together.
inelastic
(e) a snowball splats against a tree, leaving a lump of snow adhering to the bark.
9.5. Totally Inelastic Collisions
Totally inelastic collision: colliding objects stick together
maximum energy loss consistent with momentum conservation.
Pinitial m1v1 m2 v2 Pfinal m1 m2 v f
Example 9.7. 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?
Pinitial mp v p
vp
mp mc
mp
P final m p mc v c
vc
0.16 kg 6.4 kg
1.2 m / s
0.16 kg
49 m / s
Example 9.8. Fusion
2H
Consider a fusion reaction of 2 deuterium nuclei
+ 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.
Pinit mD v1 v2 Pfinal mHe v f
vf
mD
v1 v 2
mHe
2
3.5 , 0 1.8 cos 64 , sin 64 Mm / s
4
2.14 , 0.809 Mm / s
vf
2.14 0.809
2
2.3 Mm / s
2
Mm / s
tan 1
0.809
21
2.14
Example 9.9. 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.
Pinit m v Pemb m M V
Eemb
v
1
m M V 2 E final m M g h
2
V2 2 g h
Caution:
Einit
1
m v 2 E final
2
mM
V
m
v
mM
m
2gh
(heat is generated when bullet strikes block)
9.6. Elastic Collisions
Pinit m1v1i m2 v2i Pfinal m1v1 f m2 v2 f
Momentum conservation:
Energy conservation:
Einit
1
1
1
1
m1 v12i m2 v 22 i E final m1 v12 f m2 v 22 f
2
2
2
2
Implicit assumption: particles have no interaction
when they are in the initial or final states. ( 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
number of equations = 3 +1 = 4
2 more conditions needed.
( final state: v1fx , v1fy , v1fz , v2fx , v2fy , v2fz )
Elastic Collisions in 1-D
pinit m1v1i m2v2i p final m1v1 f m2v2 f
Einit
1
1
1
1
m1 v12i m2 v22 i E final m1 v12 f m2 v22 f
2
2
2
2
1-D collision
1-D case:
number of unknowns = 1 2 = 2
( v1f , v2f )
number of equations = 1 +1 = 2
This is a 2-D collision
unique solution.
m1 v1 f v1i m2 v2 f v2i
p1 p2
m1 v12f v12i m2 v22 f v22i
E1 E2
v1 f v1i v2 f v2i
v1i v2i v1 f v2 f
vi v f
m1 v1 f v1i m2 v2 f v2i
v1 f v1i v2 f v2i
v1 f
m1 m2 v1i 2m2v2i
m1 m2
(a) m1 << m2 :
m1v1 f m2v2 f m1v1i m2v2i
v1 f v2 f v1i v2i
v2 f
2m1v1i m2 m1 v2i
m1 m2
v1 f v1i 2v2i
v2i 0
(b) m1 = m2 :
v1 f v1i
v1 f v2i
v2i 0
(c) m1 >> m2 :
v2 f v2i
v1 f v1i
v2i 0
v2 f 0
v2 f v1i
v1 f 0
v2 f v1i
v2 f 2v1i v2i
v1 f v1i
v2 f 2v1i
Example 9.10. 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.
v1 f
m1 m2 v1i 2m2v2i
v2i 0
K1i
K2 f
v2 f
m1 m2
v1 f
m1 m2
v1i
m1 m2
1
v1i
3
v2 f
2m1
v1i
m1 m2
2m1v1i m2 m1 v2i
m1 m2
m1 1 u
m2 2 u
2
v1i
3
1
m1 v12i
2
1
m2 v22 f
2
K2 f
K1i
m2 v22 f
m1 v12i
4 m1 m2
m1 m2
2
4 1u 2u
1u 2u
2
8
89%
9
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 9.11. 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?
p cons:
v1i v1 f v2 f
E cons:
v12i v12f v22 f
v12i v12f 2v1 f v2 f v22 f
v12i v12f 2v1 f v2 f cos 30 v22 f
2v1 f v2 f cos 30 0
30 90
60
Center of Mass Frame
Pi Pf 0