Rotational Equilibrium and Rotational Dynamics

1
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

1. 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

1. 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

1. 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
Newtons Second Law for a Rotating Object

• How do I write Newtons second law for rotating
  rigid bodies?

17
Example, Newtons Second Law for Rotation

1. A solid, frictionless cylindrical reel of mass
   M3 kg and radius R0.4 m is used to draw water
   from a well. A bucket of mass m2 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

1. 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 R1.0x104km and an
  initial period of rotation of 30 days to R3km.
  Find the new period of rotation of the stars
  core.