Physics 103: Lecture 16 Rotational Equilibrium and Rotational Dynamics

1
Physics 103 Lecture 16 Rotational
Equilibriumand Rotational Dynamics
Moment of Inertia, Rotational Kinetic Energy,
Angular Momentum

• Reminder
• Hour Exam II, Thur, Nov. 5 545 - 7 PM
• 165 Bascom 302, 303, 304, 306, 312, 318, 320,
  324 B-10 Ingraham 305, 313, 317, 321, 322, 327, 3
  28, 329, 330
• 3650 Humanities 307, 308, 309, 310, 311, 314,
  315, 319, 323, 326 (all the same as Exam I)
• Material from Chapters 5,6,7,8 inclusive
• One page of notes (8.5 x 11) allowed
• 20 multiple choice questions plus test code
• Scantron will be used - bring 2 HB pencils
  calculator
• You must know your section number (301 - 330),
  fill it in on the test
• Alternative time signup next week. Room Cham 4320
  (the labs)

2
Inertia and Acceleration
Equilibrium forces (torques) are in balance (
zero)
Acceleration forces (torques) are non zero
Effect
Cause
How is torque related to angular
acceleration? What is the rotational equivalent
of mass? How do we express Newtons law for
rotational motion?
3
Moment of Inertia

• When a rigid object is subject to a net torque
  (?0), it undergoes an angular acceleration
• Where the force is applied on the body matters
• Distribution of mass about the body matters
• The angular acceleration is directly proportional
  to the net torque
• The relationship
• ?t Ia is analogous to ?F ma
• Newtons Second Law
• The angular acceleration is inversely
  proportional to the moment of inertia, I, of the
  object

Mass of a piece of the object (mi) Distance from
axis of rotation to that piece (ri).
SI units are kg-m2
4
Moment of Inertia of a Uniform Ring

• Image the hoop is divided into a number of small
  segments, m1
• These segments are equidistant from the axis

5
Preflight 14.5 14.6

• A hoop, a solid cylinder and a solid sphere all
  have the same mass and radius. Which of them has
  the largest moment of inertia when they rotate
  about axis shown?
• The hoop.
• The cylinder.
• The sphere
• All have the same moment of inertia

See back of lecture for formulas for inertias of
common objects
6
Preflight 14.7 14.8

• 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 x-axis?
• A.
• B.
• They have the same moment of inertia

7
Moments of Inertia
8
Rotational Kinetic Energy

• Work must be done to rotate objects
• Force expended perpendicular to the radius
• Parallel to the displacement

Ds
q
r
F
9
Angular Momentum

• Similarly to the relationship between force and
  momentum in a linear system, we can show the
  relationship between torque and angular momentum
• Angular momentum is defined as
• L Iw
• L r x p
• and torque

10
Lecture 15, Preflight 1

• The angular momentum of a particle
• is independent of the specific origin of
  coordinates.
• is zero when its position and momentum vectors
  are parallel.
• is zero when its position and momentum vectors
  are perpendicular.

Angular momentum, L I w (S mr2) (v/r) i.e., L
mv r r p (here r and p make 90o) Angular
momentum is a vector perpendicular to the
position, r, and motion, p, L r x p Right hand
rule
11
Angular Momentum and Torgue - Right Hand Rule
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Angular Momentum Conservation

• If the net torque is zero, the angular momentum
  remains constant
• Conservation of Angular Momentum states The
  angular momentum of a system is conserved when
  the net external torque acting on the systems is
  zero.
• That is, when

13
Lecture 15, Preflight 2 3
A figure skater stands on one spot on the ice
(assumed frictionless) and spins around with her
arms extended. When she pulls in her arms, she
reduces her rotational inertia and her angular
speed increases so that her angular momentum is
conserved. Compared to her initial rotational
kinetic energy, her final rotational kinetic
energy after she has pulled in her arms must be
1. Same 2. Larger because she is rotating
faster 3. Smaller because her rotational inertia
is smaller
Rotational kinetic energy is Iw2/2. LIw.
Rot.K.ELw/2 L is constant - therefore, since w
increases Rot. KE also increases. Additional
energy is provided by the skater working to
pull her arms in.
14
Total Kinetic Energy
15
Kinetic Energy Rolling without Slipping
16
Application Rolling without Slipping Down
Incline

• ?KEtotal ?PEg 0
• ?PEg -Mgh

Solve
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Application Rolling without Slipping Down
Incline
Larger I ? smaller VCM
18
Lecture 15, Preflight 4 5
Two cylinders of the same size and mass roll down
an incline. Cylinder A has most of its mass
concentrated at the rim, while cylinder B has
most of its mass concentrated at the center.
Which reaches the bottom of the incline first?
1. A 2. B 3. Both reach at the same time.
Cylinder A has higher moment of inertia than
cylinder B - therefore, it takes longer to roll
down.
19
Lecture 15, 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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Lecture 15, Preflight 6 7

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

Since there is no external torque acting on the
system, the total angular momentum is conserved.
21
Lecture 15, Preflight 8 9

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

Your moment of inertia decreases so your angular
velocity must increase to compensate for this
change and keep angular momentum the same.
22
Lecture 21, Preflight 10 11

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

Because w increases as much as I decreases. In
the equation KErot 1/2Iw2, w is squared so
the kinetic energy increases. You are doing work
by changing your moment of inertia so you
increase your kinetic energy
23
Preflight 12 13Turning 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.
• Remember, 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