Rigid Body: Rotational and Translational Motion; Rolling without Slipping 8.01 W11D1 - PowerPoint PPT Presentation

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

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


1
Rigid Body Rotational and Translational
MotionRolling without Slipping8.01W11D1
2
Todays Reading Assignment W11D1
  • Young and Freedman 10.3-10.4

3
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

4
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

5
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

6
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

7
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

8
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

9
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
10
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
11
Rolling Bicycle Wheel Constraint relations
Rolling without slipping
Rolling and Skidding
Rolling and Slipping
12
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.
13
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?
  • .
  • .
  • .
  • .

14
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,

15
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

16
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

17
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)?

18
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

19
Earths MotionSpin Angular Momentum
  • Spin angular momentum about center of mass of
    earth
  • Period and angular velocity
  • Magnitude

20
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

21
Rotational Work-Kinetic Energy Theorem
  • Change in kinetic energy of rotation about
    center-of-mass
  • Change in rotational and translational kinetic
    energy

22
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

23
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.
24
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.

25
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.

26
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?
27
Next Reading Assignment W11D1
  • Young and Freedman 10.3-10.6
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