Two-Dimensional Rotational Kinematics 8.01 W09D1 Young and Freedman: 1.10 (Vector Products) 9.1-9.6, 10.5

1
Two-Dimensional Rotational Kinematics8.01W09D1
Young and Freedman 1.10 (Vector Products)
9.1-9.6, 10.5
2
Announcements
No Math Review Night Next Week Pset 8 Due Nov 1
at 9 pm, just 3 problems W09D2 Reading
Assignment Young and Freedman 1.10 (Vector
Product) 10.1-10.2, 10.5-10.6 11.1-11.3
3
Rigid Bodies

• A rigid body is an extended object in which the
  distance between any two points in the object is
  constant in time.
• Springs or human bodies are non-rigid bodies.

4
DemoCenter of Mass and Rotational Motion of
Baton
5
Overview Rotation and Translationof Rigid Body

• Demonstration Motion of a thrown baton
• Translational motion external force of gravity
  acts on center of mass
• Rotational Motion object rotates about center
  of mass

6
Recall Translational Motion of the Center of Mass

• Total momentum of system of particles
• External force and acceleration of center of mass
  

7
Main Idea Rotational Motion about Center of Mass

• Torque produces angular acceleration about center
  of mass
• is the moment of inertial about the center
  of mass
• is the angular acceleration about center of
  mass

8
Two-Dimensional Rotational Motion

• Fixed Axis Rotation
• Disc is rotating about axis passing through the
  center of the disc and is perpendicular to the
  plane of the disc.
• Motion Where the Axis Translates
• For straight line motion, bicycle wheel rotates
  about fixed direction and center of mass is
  translating

9
Cylindrical Coordinate System

• Coordinates
• Unit vectors

10
Rotational Kinematicsfor Point-Like Particle
11
Rotational Kinematicsfor Fixed Axis Rotation

• A point like particle undergoing circular motion
  at
• a non-constant speed has
• an angular velocity vector
• (2) an angular acceleration vector

12
Fixed Axis Rotation Angular Velocity

• Angle variable
• SI unit
• Angular velocity
• SI unit
• Vector
• Component
• magnitude
• direction

13
Concept Question Angular Speed

• Object A sits at the outer edge (rim) of a
  merry-go-round, and object B sits halfway between
  the rim and the axis of rotation. The
  merry-go-round makes a complete revolution once
  every thirty seconds. The magnitude of the
  angular velocity of Object B is
• half the magnitude of the angular velocity of
  Object A .
• the same as the magnitude of the angular velocity
  of Object A .
• twice the the magnitude of the angular velocity
  of Object A .
• impossible to determine.

14
Concept Question Angular Speed

• Object A sits at the outer edge (rim) of a
  merry-go-round, and object B sits halfway between
  the rim and the axis of rotation. The
  merry-go-round makes a complete revolution once
  every thirty seconds. The magnitude of the
  angular velocity of Object B is
• Answer 2. All points in a rigid body rotate with
  the same angular velocity.

15
Example Angular Velocity

• Consider point-like object rotating with
  velocity tangent to the circle of radius r as
  shown in the figure below with
• The angular velocity vector points in the
  direction, given by

16
Fixed Axis Rotation Angular Acceleration

• Angular acceleration
• SI unit
• Vector
• Component
• Magnitude
• Direction

17
Rotational Kinematics Integral Relations
The angular quantities are exactly analogous
to the quantities for one-dimensional motion,
and obey the same type of integral relations
Example Constant angular acceleration
18
Concept Question Rotational Kinematics

• The figure shows a graph of ?z and az versus
  time for a particular rotating body. During which
  time intervals is the rotation slowing down?
• 0 lt t lt 2 s
• 2 s lt t lt 4 s
• 4 s lt t lt 6 s
• None of the intervals.
• Two of the intervals.
• Three of the intervals.

19
Concept Question Rotational Kinematics

• The figure shows a graph of ?z and az versus
  time for a particular rotating body. During which
  time intervals is the rotation slowing down?
• 0 lt t lt 2 s
• 2 s lt t lt 4 s
• 4 s lt t lt 6 s
• None of the intervals.
• Two of the intervals.
• Three of the intervals.

20
Table Problem Rotational Kinematics

• A turntable is a uniform disc of mass m and a
  radius R. The turntable is initially spinning
  clockwise when looked down on from above at a
  constant frequency f . The motor is turned off
  and the turntable slows to a stop in t seconds
  with constant angular deceleration.
• a) What is the direction and magnitude of the
  initial angular velocity of the turntable?
• b) What is the direction and magnitude of the
  angular acceleration of the turntable?
• c) What is the total angle in radians that the
  turntable spins while slowing down?

21
Summary Circular Motion for Point-like Particle

• Use plane polar coordinates circle of radius r
• Unit vectors are functions of time because
  direction changes
• Position
• Velocity
• Acceleration

22
Rigid Body Kinematicsfor Fixed Axis
RotationKinetic Energy and Moment of Inertia
23
Rigid Body Kinematicsfor Fixed Axis Rotation
Body rotates with angular velocity and
angular acceleration
24
Divide Body into Small Elements

• Body rotates with angular velocity,
• angular acceleration
• Individual elements of mass
• Radius of orbit
• Tangential velocity
• Tangential acceleration
• Radial Acceleration

25
Rotational Kinetic Energy and Moment of Inertia

• Rotational kinetic energy about axis
• passing through S
• Moment of Inertia about S
• SI Unit
• Continuous body
• Rotational Kinetic Energy

26
Discussion Moment of Inertia

• How does moment of inertia compare to the total
  mass and the center of mass?
• Different measures of the distribution of the
  mass.
• Total mass scalar
• Center of Mass vector (three components)
• Moment of Inertia about axis passing through S

27
Concept Question
All of the objects below have the same mass.
Which of the objects has the largest moment of
inertia about the axis shown?
(1) Hollow Cylinder (2) Solid Cylinder
(3)Thin-walled Hollow Cylinder
28
Concept Question
All of the objects below have the same mass.
Which of the objects has the largest moment of
inertia about the axis shown?
Answer 3. The mass distribution for the
thin-hollow walled cylinder is furthest from the
axis, hence its moment of inertia is largest.
29
Concept Question
30
Concept Question
Answer 3. The mass of the sphere is on average
closer to an axis of rotation passing through its
center then in the case of the ring or the disc
31
Concept Question
32
Concept Question
Answer 2. The mass of the ring is furthest on
average from an axis passing perpendicular to the
plane of the disc and passing through a point on
the edge of the disc.
33
Worked Example Moment of Inertia for Uniform Disc

• Consider a thin uniform disc of radius R and
  mass m. What is the moment of inertia about an
  axis that pass perpendicular through the center
  of the disc?

34
Strategy Calculating Moment of Inertia

• Step 1 Identify the axis of rotation
• Step 2 Choose a coordinate system
• Step 3 Identify the infinitesimal mass element
  dm.
• Step 4 Identify the radius, , of the
  circular orbit of the infinitesimal mass element
  dm.
• Step 5 Set up the limits for the integral over
  the body in terms of the physical dimensions of
  the rigid body.
• Step 6 Explicitly calculate the integrals.

35
Worked Example Moment of Inertia of a Disc

• Consider a thin uniform disc of radius R and
  mass m. What is the moment of inertia about an
  axis that pass perpendicular through the center
  of the disc?

36
Parallel Axis Theorem

• Rigid body of mass m.
• Moment of inertia about axis through
  center of mass of the body.
• Moment of inertia about parallel axis
  through point S in body.
• dS,cm perpendicular distance between two
  parallel axes.

37
Table Problem Moment of Inertia of a Rod

• Consider a thin uniform rod of length L and mass
  M.
• Odd Tables Calculate the moment of inertia
  about an axis that passes perpendicular through
  the center of mass of the rod.
• Even Tables Calculate the moment of inertia
  about an axis that passes perpendicular through
  the end of the rod.

38
Summary Moment of Inertia

• Moment of Inertia about S
• Examples Let S be the center of mass
• rod of length l and mass m
• disc of radius R and mass m
• Parallel Axis theorem

39
Table Problem Kinetic Energy of Disk
A disk with mass M and radius R is spinning with
angular speed ? about an axis that passes through
the rim of the disk perpendicular to its plane.
The moment of inertia about the cm is (1/2)M R2.
What is the kinetic energy of the disk?
40
Concept Question Kinetic Energy
A disk with mass M and radius R is spinning with
angular speed ? about an axis that passes through
the rim of the disk perpendicular to its plane.
Moment of inertia about cm is (1/2)M R2. Its
total kinetic energy is

• (1/4)M R2 ?2
• (1/2)M R2 ?2
• 3. (3/4)M R2 ?2

4. (1/4)M R?2 5. (1/2)M R?2 6. (1/4)M R?
41
Concept Question Kinetic Energy
Answer 3. The parallel axis theorem states the
moment of inertia about an axis passing
perpendicular to the plane of the disc and
passing through a point on the edge of the disc
is equal to The moment of inertia about an axis
passing perpendicular to the plane of the disc
and passing through the center of mass of the
disc is equal to Therefore The kinetic
energy is then

42
Summary Fixed Axis Rotation Kinematics

• Angle variable
• Angular velocity
• Angular acceleration
• Mass element
• Radius of orbit
• Moment of inertia
• Parallel Axis Theorem

43
Table Problem Pulley System and Energy

• Using energy techniques, calculate the speed of
  block 2 as a function of distance that it moves
  down the inclined plane using energy techniques.
  Let IP denote the moment of inertia of the pulley
  about its center of mass. Assume there are no
  energy losses due to friction and that the rope
  does slip around the pulley.