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1

Chapter 8

Rotational Equilibrium and Rotational Dynamics

1. Torque

2. Torque and Equilibrium

3. Center of Mass and Center of Gravity

4. Torque and angular acceleration

5. Rotational Kinetic energy

6. Angular momentum

7. 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

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

1. 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

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 System1. 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.

a) Compute the angular accelerationb) Compute the retarding torque due to frictionc) Compute the total energy transferred from mechanical

energy to thermal energy by frictiond) 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

1. 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

1. 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.

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