Transcript Lecture4

Lecture #4 of 25
Review HW problem 1.9
Explain 3 in-class problems from Tuesday
Angular Momentum


And torque
And Central force
Moment of Inertia – Worked example
DVD Demonstration on momentum cons. and
CM motion
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Impulse II -- Problem #L2-3
“Another car crash”
James and Joan are partly recovered from their previous
injuries, and haven’t learned from their experience.
They are drinking Jack Daniels and not wearing seatbelts.
James’ vehicle has velocity vector 30 xˆ m / s
Joan’s vehicle has 30 xˆ m / s
Both vehicles’ mass=M. Both people’s mass=70 kg.
Solve for case of inelastic and elastic collisions of vehicles.
Joan has an airbag in her vehicle. It takes her 100 millisec
to reduce her velocity relative to her vehicle to zero.
James stops 5 millisec after impacting the steering wheel.
What impulse does each person experience? What is
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average force for each? How many “g’s” do they feel.
Worked Example L3-1 – Discrete masses
Given m1 to m10
2 units
y
O2
y
x
O1
x
ma= m
ma = 3m
Calculate
1 unit
N
RCM 
N
ma ra  ma ra

a
a
1
N
ma

a
1

1
M total
RCM
Given origin O1
For homework
given O2
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Worked Example L3-2 – Continuous mass
2 km
Given quarter circle with
uniform mass-density s
and radius 2 km:

j
O1


r

Calculate M total
Write r in polar coords
Write out double integral,
both r and phi
components
Solve integral
Calculate
R CM
rdm  rs (r )dA



M total
M total
RCM
Given origin O1
dA  (rdj )dr
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Equivalence of Linear and Angular motion
equations
Fext
dPtotal

dt
Ptotal  M cm vcm
 ext
dLtotal

dt
Ltotal  I 
L  r  p;   r  F
vtangential    r
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Moment of Inertia vs. Center of Mass
For a multi-particle discrete mass-distribution
N
N
I   ma ra ra
R CM 
a 1
ma ra

a
1
M total
For a continuous mass-distribution
2
2
2
I   r dm   r  dV   r s dA
.
R CM
rdm  r  (r )dV  rs (r )dA




M total
M total
M total
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Angular Momentum and Torque
dL d
Lrp
 (r  p )
dt dt
d
(r  p)  (r  p)  (r  p)
dt
p  mr  (r  p )  (r  mr )  0
d
(r  p)  (r  p)  (r  F )
dt
dL
 (r  F )  
dt
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Parallel Axis Theorem
d
Axis 2 (Parallel to
axis 1)
CM
Axis 1 (through CM)
I Parallel  I CM  M total d 2
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Angular Momentum and Central Forces
Fcentral  F ( r )rˆ
dL
 (r  F )  (r  F (r ) rˆ)  0 because r  rˆ  0
dt
L  constant : Angular momentum conserved by Central forces
Table of
Position vectors
System
r
xxˆ  yyˆ  zzˆ
rrˆ  jjˆ  zzˆ
rrˆ  jjˆ  ˆ
Cartesian
Cylindrical
Spherical
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Moment of Inertia worked problem
2
L
1
Calculate the moment
of inertia and kinetic
energy of a wire of
uniform mass-density
lambda, mass M, and
length L.

A) If rotated about axis
at midpoint at angular
velocity

1
B) If rotated about axis
at endpoint at angular
velocity
2
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Lecture #4 Wind-up

2
. I  r dm
. dL
system
  external
.
dt
I Parallel  I CM  M total d 2
We are done w/ chapters 1 and 3, read all.
Assignment includes Taylor /Supplement /
Lecture probs.
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