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 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 http://www.youtube.com/watch?v=w8Tfa5fHr3s

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 r2 in them.

Momentum

Symbol

Definition

Equation

Conservation

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

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 Tail rotor failure #2

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=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)

Do cats violate physical law?

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 Gyrowheel