Rotational Equilibrium and Dynamics Rotational Inertia and Angular Momentum

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Inertia vs. Rotational Inertia Inertia is the resistance of an object to changes in its state of motion. – Objects in motion stay in motion and objects at rest stay at rest unless acted upon by unbalanced forces. Rotational Inertia is the resistance of an object to changes in its rotational state of motion. – Rotating objects tend to keep rotation and non-rotating objects tend to stay non-rotating unless acted upon by unbalanced torques. (Note: Rotational inertia is often called moment of inertia.)

Inertia vs. Rotational Inertia Recall that inertia depends on the mass of the object. – i.e. massive objects have a lot of inertia, whereas objects with small mass have little inertia. Like inertia in the linear sense, rotational inertia depends on the mass of the object, however, rotational inertia is unique in that it also depends on the distribution of the mass. (By distribution of the mass, we mean where the mass is concentrated in the object.)

Rotational Inertia Rotational inertia depends on: Total mass of the object Distribution of the mass The farther the mass is from the axis of rotation, the larger the rotational inertia or the harder it is to rotate. Rotational inertia ( I) is proportional to (mass) x (distance) 2  mr 2 (What are the units of rotational Inertia?)

Rotational Inertia Rotational Inertia is different for different objects. Some of the common rotational inertias are shown here. (see p. 285 for a more complete list)

Demo: Inertia Sticks

Demo: Hammer Balance When is the hammer easier to balance on your finger? B A

Demo: Drop the Stick Two meter sticks stand upright against a wall; one has a mass on the end. Which stick will swing down and hit the floor first? Why?

Demo: Hoop & Disk Racing Roll a hoop and a disk down a ramp; which wins the race?

Check Yourself: Why do tightrope walkers carry a long pole when walking across the high wire?

Angular Momentum As we have just discovered, rotating objects have inertia. Because of this, rotating objects also possess momentum associated with its rotation. There are two types of momentum: – Linear momentum: momentum associated with an objects linear motion p=mv (linear momentum)=(mass)x(velocity) – Angular momentum: momentum associated with an objects rotational motion L=  (angular momentum)=(moment of inertia)x(angular speed) (Notice the form of the equations is the same)

Angular Momentum L=   is the Greek letter omega and represents the angular speed of the rotating object. Angular speed is the ratio of the angular displacement (in radians) to the time.  t where the angular displacement is measured in radians. (what are the units of angular speed?) (what are the units of angular momentum?)

Calculating Angular Speed A young boy swings a yo-yo around his head such that it makes three revolutions every second. What is the angular speed of the yo-yo?

Angular Speed v. Linear Speed Angular speed is related to linear speed by the following equation: – v t =r    tangential speed)=(distance from axis)x(angular speed)

Angular Speed v. Linear Speed Tangential speed varies with the distance from the axis, angular speed does not.

Conservation of Angular Momentum: Recall that linear momentum is conserved if the net external force acting on an object or system of objects is zero. Similarly, angular momentum is conserved if the net external torque acting on an object or system of objects is zero.

Demo: Conservation of Angular Momentum

Demo: Bicycle gyro Zero Rotation Clockwise Rotation Counter- Clockwise Rotation Similar to recoil

Sample Problem A 65 kg student is spinning on a merry-go- round that has a mass of 5.25x10 2 kg and a radius of 2.00 m. She walks from the edge of the merry-go-round toward the center. If the angular speed of the merry-go-round is initially 0.20 rad/s, what is its angular speed when the student reaches a point 0.50 m from the center?

Think about it: Racing identical pop cans. What happens?