Physics 101: Chapter 9

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Physics 101 Chapter 9

• Todays lecture will cover Textbook Sections 9.1
  - 9.6

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See text chapter 8
Rotation Summary (with comparison to 1-D
kinematics)
Angular Linear
See Table 8.1
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See text chapter 9
New concept Torque
Rotational analog of force Torque (magnitude
of force) x (lever arm) t F l
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Comment on axes and sign(i.e. what is positive
and negative)

• Whenever we talk about rotation, it is implied
  that there is a rotation axis.
• This is usually called the z axis (we usually
  omit the z subscript for simplicity).
• Counter-clockwise (increasing q) is
  usuallycalled positive.
• Clockwise (decreasing q) is usuallycalled
  negative.

w
z
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Chapter 9, Preflight

• The picture below shows three different ways of
  using a wrench to loosen a stuck nut. Assume the
  applied force F is the same in each case.
• In which of the cases is the torque on the nut
  the biggest?
• 1. Case 1 2. Case 2 3. Case 3

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Chapter 9, Preflight

• The picture below shows three different ways of
  using a wrench to loosen a stuck nut. Assume the
  applied force F is the same in each case.
• In which of the cases is the torque on the nut
  the smallest?
• 1. Case 1 2. Case 2 3. Case 3

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Static Equilibrium

• A system is in static equilibrium if and only if
• acm 0 ? Fext 0
• ? 0 ? ?ext 0 (about any axis)

torque about pivot due to gravity ?g
mgd (gravity acts at center of mass)
This object is NOT in static equilibrium
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Not in equilibrium
Equilibrium
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Homework Hints

• Painter is standing to the right of the support B.

FA
FB
Mg
mg

• What is the maximum distance the painter can move
  to the right without tipping the board off?

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Homework Hints

• If its just balancing on B, then FA 0
• the only forces on the beam are

FB
x
Mg
mg
Using FTOT 0 FB Mg mg This does not
tell us x
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Homework Hints

• Find net torque around pivot B (or any other
  place)

FB
d1
d2
Mg
mg
t (FB ) 0 since lever arm is 0
t (Mg ) Mgd1
Total torque 0 Mgd1 -mgd2
t (mg ) -mgd2
So d2 Md1 /m and you can use d1 to find x
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Homework Hints

• Painter standing at the support B.

Find total torqueabout this axis
D
FA
FB
d
Mg
mg
t(FA) - FAD
t(Mg) Mgd
Total torque 0 Mgd -FAD
t(FB) 0 (since distance is 0)
So FA Mgd /D
t(mg) 0 (since distance is 0)
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• MORE EXAMPLES (bar and weights suspended by the
  string)Find net torque around this (or any
  other) place

t (m1g) 0 since lever arm is 0
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L/2
t (m1g) 0 since lever arm is 0
t (Mg ) -Mg L/2
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x
t (m1g) 0 since lever arm is 0
t (Mg ) -Mg L/2
t (T ) T x
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L
t (m1g) 0 since lever arm is 0
t (Mg ) -Mg L/2
t (T ) T x
t (m2g ) -m2g L
All torques sum to 0 Tx MgL/2 m2gL So
x (MgL/2 m2gL) / T
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Moment of Inertia Rotational KE

• Textbook Sections 9.4 - 9.5

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Torque and Stability
Center of mass over base --gt stable
Center of mass outside of base --gt unstable
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Moments of Inertia of Common Objects
Hollow cylinder or hoop about central axis I
MR2 Solid cylinder or disk about central axis I
MR2/2 Solid sphere about center I
2MR2/5 Uniform rod about center I
ML2/12 Uniform rod about end I ML2/3
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Chapter 9, Preflight

• The picture below shows two different dumbbell
  shaped objects. Object A has two balls of mass m
  separated by a distance 2L, and object B has two
  balls of mass 2m separated by a distance L.
  Which of the objects has the largest moment of
  inertia for rotations around the x-axis?
• 1. A 2. B 3. Same

m
2m
2L
L
x
2m
m
B
A
I mL2 mL2 2mL2
I 2m(L/2)2 2m(L/2)2 mL2
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Rotational Kinetic Energy
Translational kinetic energy KEtrnas 1/2
MV2cm Rotational kinetic energy KErot 1/2
I?2 Rotation plus translation KEtotal
KEtrans KErot 1/2 MV2cm 1/2 I?2
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Angular Momentum

• Textbook Section 9.6

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Define Angular Momentum
See text chapters 8-9
Momentum Angular Momentum p mV L
I? conserved if ?Fext 0 conserved if ??ext
0 Vector Vector! units kg-m/s units
kg-m2/s
See Table 8.1
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Chapter 9, Pre-flights

• You are sitting on a freely rotating bar-stool
  with your arms stretched out and a heavy glass
  mug in each hand. Your friend gives you a twist
  and you start rotating around a vertical axis
  though the center of the stool. You can assume
  that the bearing the stool turns on is
  frictionless, and that there is no net external
  torque present once you have started spinning.
• You now pull your arms and hands (and mugs) close
  to your body.

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Chapter 9, Preflight

• What happens to your angular momentum as you pull
  in your arms?
• 1. it increases 2. it decreases 3. it stays the
  same

This is like the spinning skater example in the
book. Since the net external torque is zero (the
movement of the arms and hands involve internal
torques), the angular momentum does not change.
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Chapter 9, Preflight

• What happens to your angular velocity as you pull
  in your arms?
• 1. it increases 2. it decreases 3. it stays the
  same

as with the skater example given in the
book....as you pull your arms in toward the
rotational axis, the moment of inertia decreases,
and the angular velocity increases.
My friends and I spent a good half hour doing
this once, and I can say...based on a great deal
of nausea, that the angular velocity does
increase.
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Chapter 9, Preflight

• What happens to your kinetic energy as you pull
  in your arms?
• 1. it increases 2. it decreases 3. it stays the
  same

Your angular velocity increases and moment of
inertia decreases, but angular velocity is
squared, so KE will increase with increasing
angular velocity
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Spinning disks

• Two different spinning disks have the same
  angular momentum, but disk 2 has a larger moment
  of inertia than disk 1.
• Which one has the biggest kinetic energy ?

(a) disk 1 (b) disk 2
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Preflights Turning the bike wheel

• A student sits on a barstool holding a bike
  wheel. The wheel is initially spinning CCW in
  the horizontal plane (as viewed from above). She
  now turns the bike wheel over. What happens?
• 1. She starts to spin CCW.2. She starts to spin
  CW.3. Nothing

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Turning the bike wheel...

• Since there is no net external torque acting on
  the student-stool system, angular momentum is
  conserved.
• Remenber, L has a direction as well as a
  magnitude!
• Initially LINI LW,I
• Finally LFIN LW,F LS

LS
LW,I
LW,I LW,F LS
LW,F
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Rotation Summary (with comparison to 1-d linear
motion)
See text chapters 8-9

• Angular Linear

See Table 8.1