1. Two-Dimensional Rotational Kinematics 8.01 W09D1

2. Today’s Reading Assignment: W09D1
Young and Freedman: 1.10 (Vector Products) , 10.5

3. Rigid Bodies Springs or human bodies are non-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. Demo Center of Mass and Rotational Motion of Baton

5. Overview: Rotation and Translation of 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 6

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.
Plane of Motion is Fixed:
For straight line motion, bicycle wheel rotates about fixed direction and center of mass is translating
Hyphen added in title 8

9. Cylindrical Coordinate System
Unit vectors

10. Rotational Kinematics for Point-Like Particle

11. Rotational Kinematics for 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
Vector:
Component
magnitude
direction 12

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. 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

15. Fixed Axis Rotation: Angular Acceleration
SI unit
Vector:
Component:
Magnitude:
Direction:
Address font size issue on subscript in figure 15

16. 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
Second slide for constant example 16

17. Concept Question: Rotational Kinematics
The figure shows a graph of ωz and αz versus time for a particular rotating body. During which time intervals is the rotation slowing down?
0 < t < 2 s
2 s < t < 4 s
4 s < t < 6 s
None of the intervals.
Two of the intervals.
Three of the intervals.

18. 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?

19. 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

20. Rigid Body Kinematics for Fixed Axis Rotation Kinetic Energy and Moment of Inertia

21. Rigid Body Kinematics for Fixed Axis Rotation
Body rotates with angular velocity and angular acceleration

22. 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
For accuracy and agreement with the notes, make the last item the magnitude of the radial acceleration. 22

23. 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:
Resized parentheses on unit; in general, having the superscripts extend above the delimiter looks bad. 23

24. 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: (nine possible moments)
Why nine? There’s no way I can find nine unless the integrand allows “dm x y” or something similar, which is part of an outer product, which is a tensor, and I know we agreed to stay away from these beasts. 24

25. 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 25

26. Concept Question

27. Concept Question

28. 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?

29. 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.
“dm” in text changed to italics. I’m glad to see the use of Times Roman instead of sans-serif. 29

30. 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?
We lose students here 30

31. 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. 31

32. 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.
Shouldn’t it be “passes perpendicular to the rod through its center of mass”? Do this at board

33. 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:

34. Table Problem: Kinetic Energy of Disk
A disk with mass M and radius R is spinning with angular speed ω 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?

35. Concept Question: Kinetic Energy
A disk with mass M and radius R is spinning with angular speed ω 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 ω2
(1/2)M R2 ω2
3. (3/4)M R2 ω2
4. (1/4)M Rω2
5. (1/2)M Rω2
6. (1/4)M Rω

36. Summary: Fixed Axis Rotation Kinematics
Angle variable
Angular velocity
Angular acceleration
Mass element
Radius of orbit
Moment of inertia
Parallel Axis Theorem

37. Simple Pendulum
Simple Pendulum: bob of mass m hanging from end of massless string of length I pivoted at S.
Angular velocity
Angular acceleration
Kinetic energy of rotation

38. Worked Example: Simple Pendulum
A simple pendulum is released from rest at an angle . Find the angular speed at angle

39. Worked Example: Simple Pendulum Energy Diagram
The energy diagrams for the pendulum are shown in the figure below

40. Worked Example: Simple Pendulum
Velocity
Kinetic energy
Initial energy
Final energy
Conservation of energy

41. 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.

42. Next Reading Assignment: W09D2
Young and Freedman: 1.10 (Vector Product) , ;