1. Experiment 5: Rotational Dynamics and Angular Momentum 8.01 W10D1

2. Today’s Reading Assignment: W10D1
Young and Freedman: ;
Experiment 5: Rotational Dynamics

3. Rotor Moment of Inertia

4. Review Table Problem: Moment of Inertia Wheel
A steel washer is mounted on a cylindrical rotor . The inner radius of the washer is R. A massless string, with an object of mass m attached to the other end, is wrapped around the side of the rotor and passes over a massless pulley. Assume that there is a constant frictional torque about the axis of the rotor. The object is released and falls. As the mass falls, the rotor undergoes an angular acceleration of magnitude a1. After the string detaches from the rotor, the rotor coasts to a stop with an angular acceleration of magnitude a2. Let g denote the gravitational constant.
What is the moment of inertia of the rotor assembly (including the washer) about the rotation axis?

5. Review Solution: Moment of Inertia of Rotor
Force and rotational equations while weight is descending:
Constraint:
Rotational equation while slowing down
Solve for moment of inertia:
Speeding up
Slowing down

6. Review Worked Example: Change in Rotational Energy and Work
While the rotor is slowing down, use work-energy techniques to find frictional torque on the rotor.

7. Experiment 5: Rotational Dynamics

8. Review: Cross Product
Magnitude: equal to the area of the parallelogram defined by the two vectors
Direction: determined by
the Right-Hand-Rule

9. Angular Momentum of a Point Particle
Point particle of mass m moving with a velocity
Momentum
Fix a point S
Vector from the point S to the location of the object
Angular momentum about the point S
SI Unit

10. Cross Product: Angular Momentum of a Point Particle
Magnitude:
moment arm
perpendicular momentum

11. Angular Momentum of a Point Particle: Direction
Direction: Right Hand Rule

12. Concept Question:
In the above situation where a particle is moving in the x-y plane
with a constant velocity, the magnitude of the angular momentum
about the point S (the origin)
decreases then increases
increases then decreases
is constant
is zero because this is not circular motion

13. Table Problem: Angular Momentum and Cross Product
A particle of mass m = 2 kg moves with a uniform velocity
At time t, the position vector of the particle with respect ot the point S is
Find the direction and the magnitude of the angular momentum about the origin, (the point S) at time t.

14. Solution: Angular Momentum and Cross Product
The angular momentum vector of the particle about the point S is given by:
The direction is in the negative direction, and the magnitude is

15. Angular Momentum and Circular Motion of a Point Particle:
Fixed axis of rotation: z-axis
Angular velocity
Velocity
Angular momentum about the point S

16. Concept Question
A particle of mass m moves in a circle of radius R at an angular speed ω about the z axis in a plane parallel to but above the x-y plane. Relative to the origin
is constant.
is constant but is not.
is constant but is not.
has no z-component. .

17. Worked Example : Changing Direction of Angular Momentum
A particle of mass m moves in a circle of radius R at an angular speed ω about the z axis in a plane parallel to but a distance h above the x-y plane.
Find the magnitude and the direction of the angular momentum relative to the origin.
Also find the z component of

18. Solution: Changing Direction of Angular Momentum

19. Solution: Changing Direction of Angular Momentum

20. Table Problem: Angular Momentum of a Two Particles About Different Points
Two point like particles 1 and 2, each of mass m, are rotating at a constant angular speed about point A. How does the angular momentum about the point B compare to the angular momentum about point A? What about at a later time when the particles have rotated by 90 degrees?

21. Table Problem: Angular Momentum of Two Particles
Two identical particles of mass m move in a circle of radius R, 180º out of phase at an angular speed ω about the z axis in a plane parallel to but a distance h above the x-y plane.
a) Find the magnitude and the direction of the angular momentum relative to the origin.
b) Is this angular momentum relative to the origin constant? If yes, why? If no, why is it not constant?

23. Table Problem: Angular Momentum of a Ring
A circular ring of radius R and mass dm rotates at an angular speed ω about the z-axis in a plane parallel to but a distance h above the x-y plane.
a) Find the magnitude and the direction of the angular momentum relative to the origin.
b) Is this angular momentum relative to the origin constant? If yes, why? If no, why is it not constant?

25. Concept Question
A non-symmetric body rotates with constant angular speed ω about the z axis. Relative to the origin
is constant.
is constant but is not.
is constant but is not.
has no z-component.

26. Concept Question
A rigid body with rotational symmetry body rotates at a constant angular speed ω about it symmetry (z) axis. In this case
is constant.
is constant but is not.
is constant but is not.
has no z-component.
5. Two of the above are true.

27. Angular Momentum of Cylindrically Symmetric Body
A cylindrically symmetric rigid body rotating about its symmetry axis at a constant angular velocity with has angular momentum about any point on its axis

28. Angular Momentum for Fixed Axis Rotation
Angular Momentum about the point S
Tangential component of momentum
z-component of angular momentum
about S:

29. Concept Question: Angular Momentum 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 magnitude of its angular momentum is:

30. Kinetic Energy of Cylindrically Symmetric Body
A cylindrically symmetric rigid body with moment of inertia Iz rotating about its
symmetry axis at a constant angular velocity
Angular Momentum
Kinetic energy

31. Concept Question: Figure Skater
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 moment of inertia and her angular speed increases. Assume that her angular momentum is constant. Compared to her initial rotational kinetic energy, her rotational kinetic energy after she has pulled in her arms must be
the same.
larger.
smaller.
not enough information is given to decide.

32. Next Reading Assignment: W10D2
Young and Freedman: ;
Experiment 6: Conservation of Angular Momentum