Rotational Inertia and Angular Momentum

Inertia

• The resistance of an object to change its state of motion

• Depends on mass (the bigger the mass, the bigger the inertia).

Rotational Inertia

• The resistance of an object to change its state of rotation

• Depends on the distribution of mass: the further the mass is from the axis of rotation, the more rotational inertia

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Why do some racing bicycles have solid wheels?

• Which has more rotational inertia, a spoke wheel or a solid wheel?

• Why?

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Which has more rotational inertia?

• Spoke wheel has more rotational inertia because mass is distributed farther from axis of rotation.

• Harder to start and stop rotating than solid wheels.

Rotational Inertia Application

• Tight rope walkers typically carry long poles with lots of mass on the ends.

• Why?

Rotational Inertia Wrap Up

• Mass is further from the axis of rotation.

• An object will have _______ rotational inertia.

• It will be _______ to rotate.

• It will be _______ to balance.

• Mass is closer to the axis of rotation.

• An object will have _______ rotational inertia.

• It will be _______ to rotate.

• It will be _______ to balance.

Force vs. Torque

• Force causes things to accelerate.

• If you want something to accelerate, apply a force.

• F = ma• Units: Newtons (N)

• Torque causes things to rotate.

• If you want something to rotate, apply a torque.

• Torque = perpendicular force x lever arm

• Units: mN or Nm

Torque

torque = force x lever armperpendicular part of forcedistance from wherethe force is applied to the axis of rotation

force applied herelever arm

For example: Let’s say you are opening a door.

Loosening a bolt

• To get a bolt to turn you must apply a torque to it.

• Describe a few different ways force a bolt to turn.

Some torque more torque,

____________

Even more torque,

____________

Torque & dragsters

• In terms of torque, why do dragsters have really big back tires?

Torque and Balancing

• If a teeter-totter is balanced, that means that the torque on both sides is equal but in the opposite direction.

• A common mistake is that students think the force is the same on both sides but that is not true. THE TORQUE MUST BE THE EQUAL!

Torque Practice Problem

• A 40 kg child sits 1.5 meters away from the center of a teeter-totter. How much force does his dad need to exert 2 meters from the center (on the other side) to prevent the child from moving?

40 kgF = ?

Angular Momentum

• Linear Momentum = mass x velocity ( p = mv )

• Angular Momentum = rotational inertia x angular

speed

Conservation of Angular Momentum

Why do ice skaters spin so much faster when they bring their arms close to their bodies?

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

QuickTime™ and aMicrosoft Video 1 decompressorare needed to see this picture.

Conservation of Angular Momentum

To understand why ice skaters, divers, & gymnasts spin faster when they bring their bodies in we must discuss the Conservation of Angular Momentum.

Law of Conservation of Angular Momentum

• If no external torque acts on a rotating system, the angular momentum of that system is constant.

Angular Momentum Before = Angular Momentum After

Rot. Inertia x Rot. Velocity = Rot. Inertia x Rot. Velocity

(before) (after)

The Angular Momentum for both is the same because they have to be the same…IT’S A LAW!

big rotational inertia small rotational inertia

small rotational velocity big rotational velocity

Angular Momentum and Vectors

Angular Momentum like linear momentum is a vector.

• What does that mean?

Direction is important!

The Direction of Angular Momentum

Because the direction of something rotating is hard to determine, physicists say that the direction of angular momentum is in the plane of the rotation.

If this wheel was rotating, we would say its angular momentum is pointed in this direction.

So it would want to stay rotating in that direction.

Examples of Angular Momentum

So next time you play:

Remember you are just conserving angular momentum.

- Football

- Frisbee

- Ride your bike