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ROTATIONAL INERTIA & ANGULAR MOMENTUM
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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)
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Rotational Inertia (angular equivalent) Symbol Definition Limitations Depends on II 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)
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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
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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
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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 http://www.youtube.com/watch?v=w8Tfa5fHr3s http://www.youtube.com/watch?v=w8Tfa5fHr3s
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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
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Other Examples: Spinning in zero Gravity Splash! Time Warp: Optimal Dive
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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.
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Momentum Symbol Definition Equation Conservation pp 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
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Angular Momentum Symbol Definition Equation Conservation LL 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
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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
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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:
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Examples: Helicopter tail rotor failure Helicopter tail rotor failure Tail rotor failure #2 Tail rotor failure #2
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Sports Connection… Ice skating Skater starts out in slow spin with arms and legs out http://www.youtube.com/watch?v=AQLtcEAG9v0 http://www.youtube.com/watch?v=AQLtcEAG9v0 http://www.youtube.com/watch?v=NtEnEeEyw_s http://www.youtube.com/watch?v=NtEnEeEyw_s 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)
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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
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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
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Applications… The Big Cheese! The Big Cheese! The Gyrowheel The Gyrowheel
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