Chapter 8Rotational Equilibrium and Rotational Dynamics

1. Torque2. Torque and Equilibrium3. Center of Mass and Center of Gravity4. Torque and angular acceleration5. Rotational Kinetic energy6. Angular momentum7. Conservation of angular momentum

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?

Torque

But wait, what does the torque equation really mean?

Lever Arm

What is a lever arm? How does it help?

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

Right Hand Rule1. 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?

Torque and Equilibrium

Example - Equilibrium1. 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.

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

Center of Gravity What is center of

gravity? How do I calculate

it? Is there an easier

way? What about

arbitrary objects?

Example - Center of Gravity

1. Find the center of gravity for the 3 mass system shown in the figure.

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

Moment of Inertia of a Uniform Ring

Other Moments of Inertia

Torque and Angular Acceleration

Newton’s Second Law for a Rotating Object How do I write Newton’s second

law for rotating rigid bodies?

Example, Newton’s Second Law for Rotation1. 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?

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

Total Energy of a System Conservation of Mechanical Energy

Example - Rotational Kinetic Energy

1. A sphere and a cylinder rolls down an inclined plane of height h. Which object reaches the bottom first?

Work-Energy in a Rotating System

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.

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?

Example - Angular Momentum1. 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.

Example - Angular Momentum: Neutron Star1. 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.