Rigid Body: Rotational and Translational Motion; Rolling without Slipping 8.01 W11D1

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Rigid Body Rotational and Translational
MotionRolling without Slipping8.01W11D1
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Todays Reading Assignment W11D1

• Young and Freedman 10.3-10.4

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Overview Rotation and Translation of Rigid Body

• Thrown Rigid Rod
• Translational Motion the gravitational
  external force acts on center-of-mass
• Rotational Motion object rotates about
  center-of-mass. Note that the center-of-mass may
  be accelerating

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Overview Rotation about the Center-of-Mass of a
Rigid Body

• The total external torque produces an angular
  acceleration about the center-of-mass
• is the moment of inertial about the
  center-of-mass
• is the angular acceleration about the
  center-of-mass
• is the angular momentum about the
  center-of-mass

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Fixed Axis Rotation

• CD is rotating about axis passing through the
  center of the disc and is perpendicular to the
  plane of the disc.
• For straight line motion, bicycle wheel rotates
  about fixed direction and center of mass is
  translating

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Review Relatively Inertial Reference Frames

• Two reference frames.
• Origins need not coincide.
• One moving object has different
• position vectors in different frames
• Relative velocity between the two reference
  frames
• is constant since the relative acceleration is
  zero

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Review Law of Addition of Velocities

• Suppose the object is moving then, observers in
  different reference frames will measure different
  velocities
• Velocity of the object in Frame 1
• Velocity of the object in Frame 2
• Velocity of an object in two different reference
  frames

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Center of Mass Reference Frame

• Frame O At rest with respect to ground
• Frame Ocm Origin located at center of mass
• Position vectors in different frames
• Relative velocity between the two reference
  frames
• Law of addition of velocities

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Rolling Bicycle Wheel
Reference frame fixed to ground
Center of mass reference frame
Motion of point P on rim of rolling bicycle
wheel Relative velocity of point P on rim
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Rolling Bicycle Wheel
Distance traveled in center of mass reference
frame of point P on rim in time ?t
Distance traveled in ground fixed reference frame
of point P on rim in time ?t
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Rolling Bicycle Wheel Constraint relations
Rolling without slipping
Rolling and Skidding
Rolling and Slipping
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Rolling Without Slipping velocity of points on
the rim in reference frame fixed to ground
The velocity of the point on the rim that is in
contact with the ground is zero in the reference
frame fixed to the ground.
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Concept Question Rolling Without Slipping

• When the wheel is rolling without slipping what
  is the relation between the final center-of-mass
  velocity and the final angular velocity?
• .
• .
• .
• .

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Angular Momentum for 2-Dim Rotation and
Translation

• The angular momentum for a rotating and
  translating object is given by (see next two
  slides for details of derivation)
• The first term in the expression for angular
  momentum about S arises from treating the body as
  a point mass located at the center-of-mass moving
  with a velocity equal to the center-of-mass
  velocity,
• The second term is the angular momentum about
  the center-of mass,

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Derivation Angular Momentum for 2-Dim Rotation
and Translation

• The angular momentum for a rotating and
  translating object is given by
• The position and velocity with respect to the
  center-of-mass reference frame of each mass
  element is given by
• So the angular momentum can be expressed as

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Derivation Angular Momentum for 2-Dim Rotation
and Translation

• The two middle terms in the above expression
  vanish because in the center-of-mass frame, the
  position of the center-of-mass is at the origin,
  and the total momentum in the center-of-mass
  frame is zero,
• Then then angular momentum about S becomes
• The momentum of system is
• So the angular momentum about S is

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Table Problem Angular Momentum for Earth

• What is the ratio of the spin angular momentum
  to the orbital angular momentum of the Earth?
• What is the vector expression for the total
  angular momentum of the Earth about the center of
  its orbit around the sun (you may assume the
  orbit is circular and centered at the sun)?

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Earths Motion Orbital Angular Momentum about Sun

• Orbital angular momentum about center of sun
• Center of mass velocity and angular velocity
• Period and angular velocity
• Magnitude

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Earths MotionSpin Angular Momentum

• Spin angular momentum about center of mass of
  earth
• Period and angular velocity
• Magnitude

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Earths Motion about Sun Orbital Angular Momentum

• For a body undergoing orbital motion like the
  earth orbiting the sun, the two terms can be
  thought of as an orbital angular momentum about
  the center-of-mass of the earth-sun system,
  denoted by S,
• Spin angular momentum about center-of-mass of
  earth
• Total angular momentum about S

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Rotational Work-Kinetic Energy Theorem

• Change in kinetic energy of rotation about
  center-of-mass
• Change in rotational and translational kinetic
  energy

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Rules to Live By Kinetic Energy of Rotation and
Translation

• Change in kinetic energy of rotation about
  center-of-mass
• Change in rotational and translational kinetic
  energy

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Concept Question
Two cylinders of the same size and mass roll
down an incline, starting from rest. 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 first? 1) A 2) B 3) Both at the same
time.
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Problem Cylinder on Inclined Plane Energy Method

• A hollow cylinder of outer radius R and mass m
  with moment of inertia I cm about the center of
  mass starts from rest and moves down an incline
  tilted at an angle q from the horizontal. The
  center of mass of the cylinder has dropped a
  vertical distance h when it reaches the bottom of
  the incline. Let g denote the gravitational
  constant. The coefficient of static friction
  between the cylinder and the surface is ms. The
  cylinder rolls without slipping down the incline.
  Using energy techniques calculate the velocity of
  the center of mass of the cylinder when it
  reaches the bottom of the incline.

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Concept Question Angular Collisions

• A long narrow uniform stick lies motionless on
  ice (assume the ice provides a frictionless
  surface). The center of mass of the stick is the
  same as the geometric center (at the midpoint of
  the stick). A puck (with putty on one side)
  slides without spinning on the ice toward the
  stick, hits one end of the stick, and attaches to
  it.
• Which quantities are constant?
• Angular momentum of puck about center of mass of
  stick.
• Momentum of stick and ball.
• Angular momentum of stick and ball about any
  point.
• Mechanical energy of stick and ball.
• None of the above 1-4.
• Three of the above 1.4
• Two of the above 1-4.

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Problem Angular Collision
A long narrow uniform stick of length l and mass
m lies motionless on ice (assume the ice provides
a frictionless surface). The center of mass of
the stick is the same as the geometric center (at
the midpoint of the stick). The moment of
inertia of the stick about its center of mass is
lcm. A puck (with putty on one side) has the same
mass m as the stick. The puck slides without
spinning on the ice with a speed of v0 toward the
stick, hits one end of the stick, and attaches to
it. You may assume that the radius of the puck is
much less than the length of the stick so that
the moment of inertia of the puck about its
center of mass is negligible compared to lcm.
What is the angular velocity of the stick plus
puck after the collision? How far does the
stick's center of mass move during one rotation
of the stick?
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Next Reading Assignment W11D1

• Young and Freedman 10.3-10.6