1. Rotational Equilibrium and Rotational Dynamics
Chapter 8
Rotational Equilibrium and Rotational Dynamics
Torque
Torque and Equilibrium
Center of Mass and Center of Gravity
Torque and angular acceleration
Rotational Kinetic energy
Angular momentum
Conservation of angular momentum

2. Torque What is torque? How do I calculate it? What are its SI units?
How do is compare to force?
How do I find the direction of torque?
How do I add two or more torques?

3. Torque
But wait, what does the torque equation really mean?

4. Lever Arm
What is a lever arm?
How does it help?

5. Right Hand Rule
Point the fingers in the direction of the position vector
Curl the fingers toward the force vector
The thumb points in the direction of the torque

6. Right Hand Rule
A fishing pole is 2.00 m long and inclined to the horizontal at an angle of 20.0° (Fig. P8.6). What is the torque exerted by the fish about an axis perpendicular to the page and passing through the hand of the person holding the pole?

7. Torque and Equilibrium

8. Example - Equilibrium
A uniform horizontal 300-N beam, 5.00 m long, is attached to a wall by a pin connection that allows the beam to rotate. Its far end is supported by a cable that makes an angle of 53.0° with the horizontal. If a 600-N person stands 1.50 m from the wall, find the tension in the cable and the force exerted by the wall on the beam.

9. Axis of Rotation
If the object is in equilibrium, it does not matter where you put the axis of rotation for calculating the net torque
The location of the axis of rotation is completely arbitrary
Often the nature of the problem will suggest a convenient location for the axis
When solving a problem, you must specify an axis of rotation
Once you have chosen an axis, you must maintain that choice consistently throughout the problem

10. Center of Gravity What is center of gravity? How do I calculate it?
Is there an easier way?
What about arbitrary objects?

11. Example - Center of Gravity
Find the center of gravity for the 3 mass system shown in the figure.

12. Moment of Inertia What is moment of Inertia? How do I calculate it?
What are its SI units?

13. Moment of Inertia of a Uniform Ring

14. Other Moments of Inertia

15. Torque and Angular Acceleration

16. Newton’s Second Law for a Rotating Object
How do I write Newton’s second law for rotating rigid bodies?

17. Example, Newton’s Second Law for Rotation
A solid, frictionless cylindrical reel of mass M=3 kg and radius R=0.4 m is used to draw water from a well. A bucket of mass m=2 kg is attached to a cord that is wrapped around the cylinder. If the bucket starts from rest at the top of the well and falls for 3.0 s before hitting the water, how far does it fall?

18. Rotational Kinetic Energy
How do I calculate it?
What are the SI units?

19. Total Energy of a System
Conservation of Mechanical Energy

20. Example - Rotational Kinetic Energy
A sphere and a cylinder rolls down an inclined plane of height h. Which object reaches the bottom first?

21. Work-Energy in a Rotating System

22. Example - Work-Energy in a Rotating System
Attached to each end of a thin steel rod of length 1m and mass 6.2 kg is a small ball of mass 1.10 kg. The rod is constrained to rotate in a horizontal plane about a vertical axis through its midpoint. At a certain instant, it is rotating at 39.0 rev/s, because of friction, it slows to a stop in 32 s. Assume a constant frictional torque.
Compute the angular acceleration
Compute the retarding torque due to friction
Compute the total energy transferred from mechanical energy to thermal energy by friction
Compute the number of revolutions rotated during 32 s.

23. Angular Momentum What is angular momentum? How do I calculate it?
What are the SI units?
How do I relate it to torque?
What about conservation?

24. Example - Angular Momentum
A student sits on a rotating stool holding two 3.0-kg objects. When his arms are extended horizontally, the objects are 1.0 m from the axis of rotation, and he rotates with an angular speed of 0.75 rad/s. The moment of inertia of the student plus stool is 3.0 kg • m2 and is assumed to be constant. The student then pulls the objects horizontally to 0.30 m from the rotation axis. (a) Find the new angular speed of the student. (b) Find the kinetic energy of the student before and after the objects are pulled in.

25. Example - Angular Momentum: Neutron Star
During a supernovae explosion a stars core collapses from a radius of R=1.0x104km and an initial period of rotation of 30 days to R=3km. Find the new period of rotation of the star’s core.