Why does a diver rotate faster when she tucks in her arms and legs

1
Why does a diver rotate faster when she tucks in
her arms and legs?
Physics 1710Warm-up Quiz

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• She increases her angular momentum.
• She increases her moment of inertia.
• She decreases her moment of inertia.
• She pushes against her inertia.
• None of the above

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Physics 1710Chapter 13 App E E
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• Analysis
• Like an ice skater. Why does an ice skater
  increase her angular velocity without the benefit
  of a torque?
• L r x p r x ( m v)
• r x ( m r x ?)
• Li mi ri 2 ? z
• Lz (?i mi ri 2 ) ?z
• Lz Iz ?z ?z Lz / Iz
• Therefore, a decrease in I ( by reducing r) will
  result in an increase in ? even if dL/dt 0!

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Physics 1710Chapter 13 App E E
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• Rotating Platform Demonstration

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Physics 1710Chapter 13 App E E
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• Analysis
• Why does an ice skater increase her angular
  velocity without the benefit of a torque?
• L r x p
• r x ( m v)
• r x ( m r x ?)
• Li mi ri 2 ?
• Lz (?i mi ri 2 ) ?
• Lz I ? ? Lz / I
• Therefore, a decrease in I ( by reducing r) will
  result in an increase in ?.

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Physics 1710Chapter 13 App E E
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• How does a ladder stay up?

Peer Instruction Time
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Physics 1710Chapter 13 App E E
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• 1' Lecture
• Static equilibrium (no translational or
  rotational acceleration) requires that all forces
  and torques to balance.
• If the acceleration due to gravity is the same
  for all particles comprising a body the center of
  mass is the center of gravity. CM CG if g the
  same for all mi .
• The moduli of elasticity (Y, E, B) characterizes
  the stress-strain relation
• stress modulus strain s Y e

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Physics 1710Chapter 13 App E E
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• One more turn on LAngular Momentum

L r x p The angular momentum is the vector
product of the moment arm and the linear
momentum. ? T d L/dt The net torque is equal
to the time rate of change in the angular
momentum.
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Physics 1710Chapter 12 App E E
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• Second Law of Motion

F m a Or F dp/dt Then r x F d (r x
p)/dt Torque t d L/dt
L r x p is the angular momentum.
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Physics 1710Chapter 12 App E E
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• Second Law of Motion

Torque t d L/dt If t 0, then L is a
constant.
L constant means angular momentum is conserved.
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Physics 1710Chapter 12 App E E
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• Second Law of Motion Gyroscopic Precession

L I ?
Torque t d L/dt d( I ?)/dt
t r x F d L/dt I (d?/dt) d?/dt
I-1 t
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Physics 1710Chapter 13 App E E
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• What will happen to a tilted spinning top when it
  is supported on one end only? Why?

Peer Instruction Time
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What will happen to a tilted, spinning top when
it is supported on one end only? Why?
Physics 1710Chapter 11 App E E

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• It will fall over because of gravity.
• It will spin faster because of I is changing.
• It will precess clockwise due to torque.
• It will precess counterclockwise due to torque.
• None of the above.

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Physics 1710Chapter 11 App E E
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• Experiment

• Spinning Top
• Observe the direction of rotation (?)
• Observe the motion (d?/dt)

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Physics 1710Chapter 11 App E E
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• Second Law of Motion Gyroscopic Precession

L I ?
Torque t d L/dt d( I ?)/dt
t r x F d L/dt I (d?/dt) d?/dt
I-1 t
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Physics 1710Chapter 11 App E E
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• Torque and the Right Hand Rule

r
X
F
r x F
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Physics 1710Chapter 11 App E E
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• Gyroscopic Precession

Torque t d L/dt d( I ?)/dt
I ( d?/dt)
(Top view)
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Physics 1710Chapter 11 App E E
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• Fundamental Angular Momentum
• Fundamental unit of angular momentum ?
• ? 1.054 x 10 -34 kg?m/s2
• ICM? ?
• ? ? / ICM
• 1.054 x 10 -34 kg?m/s2 / (1.95 x 10 -46 kg?m)
• 5.41 x 10 11 rad/s

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Physics 1710Chapter 11 App E E
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• MRI (a.k.a. NMR Nuclear Magnetic Resonance)
• Magnetic Resonance Imaging

Magnetic Field
Torque?
Hydrogen atoms
Precession!

MRI permits imaging of soft tissue due precession
of the hydrogen in the water of the human body.




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Physics 1710Chapter 11 App E E
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• Teeter-totter How does it balance?

Fsupport - Fg
tnet Sti
For equilibrium tnet Sti 0 Fnet SFi 0
FgF1F2
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Physics 1710Chapter 11 App E E
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How does a ladder stay up?
Good ideas?
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Physics 1710Chapter 11 App E E
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• Legends of the FallHow a ladder stays up

Fnet 0
Tnet 0
Fnet 0
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Physics 1710Chapter 11 App E E
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• Equilibrium (equi equal libium scales
  ?)

Equilibrium implies balanced. Fnet d P/dt In
equilibrium Fnet 0 tnet d L/dt In
equilibrium tnet 0
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Can an object be in equilibrium when the center
of mass lies outside of the object?
Physics 1710Chapter 11 App E E

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• Yes.
• No.
• Depends.

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Physics 1710Chapter 11 App E E
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• Solution

In equilibrium Fnet 0 tnet 0
Hollow Cone
Is it stable? i.e. does it recover from a small
displacement ?
25
Physics 1710Chapter 11 App E E
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• Center of Gravity
• If the acceleration due to gravity is the same
  for all parts of a body, then the center of mass
  corresponds to the center of gravity. CG CM
  if g uniform
• Proof Ti Sxi Fgi Sxi gmgi Mg Sxi mgi
  /M Fcg xcm

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Physics 1710Chapter 11 App E E
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• Center of Gravity (comparison)

CM
CM
Ball Fg - mg
CG
Moon Fg - GmM/r2
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Physics 1710Chapter 11 App E E

• Elasticity
• Definitions
• Stress s the deforming force per unit area.
• Strain e the unit deformation.
• Stress modulus x strain
• s F/A Y e

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Physics 1710Chapter 11 App E E

• Elasticity
• Stress s Strain e Curve

Stress s (N/m2)
s Y e
Strain e ?L/L ()
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Physics 1710Chapter 11 App E E
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• Elasticity
• Stress s the deforming force per unit area.
• Strain e the unit deformation.
• Tensile/Compressive Stress Youngs Modulus E
• Stress modulus x strain
• s F/A E e E ?L/L

?L
L
s E e
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Physics 1710Chapter 11 App E E
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• Elasticity
• Stress s the deforming force per unit area.
• Strain e the unit deformation.
• Shear Modulus G
• Stress modulus x strain
• s F/A G e G ?x/L

L
?x
s
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Physics 1710Chapter 11 App E E
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• Elasticity
• Stress s the deforming force per unit area.
• Strain e the unit deformation.
• Hydraulic Stress Bulk Modulus B
• Stress modulus x strain
• s F/A p B e B ?V/V

?V
p
V
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Physics 1710Chapter 11 App E E

• Elasticity
• Stress the deforming force per unit area.
• Strain the unit deformation.
• Tensile stretch
• Compressive squeeze
• Shear lean
• Hydraulic pressure
• Yield permanently deformed

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Physics 1710Chapter 11 App E E

• Summary
• Static equilibrium implies that all forces and
  torques balance.
• The center of mass is often the center of
  gravity.
• The moduli of elasticity characterizes the
  stress-strain relation
• stress modulus x strain
• Stress modulus x strain
• s F/A Y e

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Physics 1710Chapter 11 App E E
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Physics 1710Chapter 11 App E E
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• Why does the platform spin faster when he brings
  his arms in?

Peer Instruction Time
36
Why does the platform spin faster when he brings
his arms in?
Physics 1710Chapter 11 App E E

• He increases his angular momentum.
• He increase his moment of inertia.
• He decrease his moment of inertia.
• He pushes against the inertia of the weights.
• None of the above

37
Where should the fulcrum be place to balance the
teeter-totter?
Physics 1710Chapter 11 App E E

38
Which way will the torque ladder move?
Physics 1710Chapter 11 App E E

• Clockwise
• Counterclockwise
• Will stay balanced