ROTATIONAL INERTIA & ANGULAR MOMENTUM

Inertia (linear quantity) Symbol Definition Limitations Depends on  m (mass)  An object at rest tends to stay at rest and an object in motion tends to stay in motion unless…  Acted upon by an outside force  Mass (more mass = more inertia)

Rotational Inertia (angular equivalent) Symbol Definition Limitations Depends on II  An object not rotating tends to stay not rotating and an object rotating about an axis tends to stay rotating about that axis unless…  Acted upon by an outside torque  Mass distribution (more mass farther from axis of rotation = more rotational inertia)

Rotational Inertia( I)  Inertia is a measure of laziness!  Resistance to the change in rotational motion  Objects that are rotating about an axis tend to stay rotating, objects not rotating tend to remain at rest, unless an outside torque is applied  A torque is required to change the status of an object’s rotation

Rotational Inertia (cont.)  Some objects have more rotational inertia than others  Objects with mass closer to axis of rotation are easier to rotate, b/c it they have less rotational inertia  If the mass is farther away from the axis, then object will have more rotational inertia, and will therefore be harder to rotate

Why does a tightrope walker carry a long pole?  The pole is usually fairly heavy and by carrying it, he creates a lot of mass far away from the axis of rotation  This increases his rotational inertia  And therefore makes it harder for him to rotate/tip over 

Sports Connection  Running  When you run you bend your legs to reduce your rotational inertia  Gymnastics/Diving  Pull body into tight ball to achieve fast rotation

Other Examples: Spinning in zero Gravity Splash! Time Warp: Optimal Dive

The big idea  Rotational Inertia depends on mass and radius  If either one of these is large, then rotational inertia is large, and object will be harder to rotate  Different types of objects have different equations for rotational inertia  But all equations have m and r 2 in them.

Momentum Symbol Definition Equation Conservation pp  Inertia in motion  Momentum = mass x velocity (p=mv)  If no unbalanced external force acts on an object, the momentum of that object is conserved

Angular Momentum Symbol Definition Equation Conservation LL  Inertia of rotation  Angular momentum = rotational inertia x rotational velocity (L = I )  If no unbalanced external torque acts on a rotating system, the angular momentum of that system is conserved

Conservation of Angular Momentum  If no outside torque is being applied, then total angular momentum in a system must stay the same  This means, if you decrease radius, you increase rotational speed  Increase radius, then rotational speed decreases I – represents rotational inertia ω -represents angular speed

Angular Momentum  The more rotational inetia has (the more mass farther out from the center) and the higher the rotational velocity, the more angular momentum it has. Example:

Examples:  Helicopter tail rotor failure Helicopter tail rotor failure  Tail rotor failure #2 Tail rotor failure #2

Sports Connection…  Ice skating  Skater starts out in slow spin with arms and legs out    Skater pulls arms and legs in tight to body  Skater is then spinning much faster (higher rotational speed)  Gymnastics (pummel horse or floor routine)  Small radius to achieve fast rotational speed during moves, increase radius when low rotational speed is desired (during landing)

Do cats violate physical law?  Video Video  They rotate their tail one way, so that their body rotates the other so that their feet are facing the ground and they land on their feet.  This combined with their flexibility allow them to almost always land on their feet 17

Universe Connection  Rotating star shrinks radius…. What happens to rotational speed??  Goes way up….. Spins very fast  Rotating star explodes outward…. What happens to rotational speed??  Goes way down … spins much slower

Applications…  The Big Cheese! The Big Cheese!  The Gyrowheel The Gyrowheel