Unit 4 Summary

Rotational Inertia/Angular Momentum

Inertia is an object's resistance to change. 

Rotational Inertia is the property of an object to resist changes in the a spin. For example, an object with a smaller amount of rotational inertia is easier to spin than an object with a larger rotational inertia.

If you remember the Conservation of Linear Momentum that states (total momentum before = total momentum after), you will remember that momentum is ALWAYS conserved. So...

The Conservation of Angular Momentum is the law that states
(angular momentum before = angular momentum after).

Angular momentum requires two things

1) rotational inertia

2) rotational velocity

So... angular momentum = (rotational inertia)(rotational velocity)

(rotational inertia)(rotational velocity) = (rotational inertia)(rotational velocity)

This is the before = after

For example 

The ice skater on the left has her arms out. This puts her arms' mass farther from her axis of rotation, giving her a larger rotational inertia and thus a smaller rotation velocity. The ice skater on the right has her arms pulled in. This puts her arms' mass closer to her axis of rotation, giving her a smaller rotational inertia and thus a larger rotational velocity. This is why ice skaters spin faster when their arms are pulled in to their chests.

Rotational & Tangential Velocities 

Same rotational-different tangential --> Train wheels are designed to keep the train on the track by having the same rotational speed, but different tangential speeds. They're designed to have narrow edges on the outside and a wider wheel on the inside. This causes the wheels to have the same rotational velocities (moving the same distance over the same period of time) because they make the same amount of rotations. When the wider wheel ends upon the far side of the track then that side will be going faster and will cause the train to curve back towards the middle (self-correcting).

Different rotational-same tangential --> Gears are designed to have the same tangential speeds. Even when a smaller gear and a larger gear are connected, they are going the same speed. Although, the smaller wheel isn't making as many rotations as the larger gear, thus giving the two gears different rotational velocities (the amount of rotations made during a period of time).

Torques

A torque is a force acting over a perpendicular distance (lever arm), and it is caused by rotation. A torque happens when an object's center of gravity goes outside of it's base of support. 

This object's center of gravity is outside it's base of support, so a torque is caused. This object will be easy to knock over.

*An object will be more likely to have a torque if its base is narrow or if it has a high center of gravity, and will be less likely to have a torque if its base is wide or if it has a low center of gravity.

This object's center of gravity is inside it's base of support, so a torque is NOT caused. This object will be hard to knock over. 

*An object will be more likely to have a torque if its base is narrow or if it has a high center of gravity, and will be less likely to have a torque if its base is wide or if it has a low center of gravity.

This system is balanced, because the torques are equal. 

Torque = (force)(lever arm)

Counter-clockwise torque = clockwise torque

(F)(lever arm) = (F)(lever arm)


*It is important to remember that...
1) force does NOT = force
2) lever arm does NOT = lever arm
3) torque = torque 

Centripetal Force 

Centripetal force is the force that makes a body follow a curved path.
*Centrifugal force (the center fleeing force) is NOT a real force. 

This ball is connected to a string attached to a ceiling. When the ball is pushed, it will continue moving in a circular path due to centripetal force, which causes the ball to rotate around its axis of rotation (where the string connects to the ceiling)

String = F tension

Ball = F weight
*The centripetal force is in the inward direction, because it will NEVER be outward.

Example: In a washing machine, there are little holes on the inside metal to let the water drain out of the clothes. The clothes have a centripetal force in addition to their tangential velocity. The water only has a tangential velocity, because it is small enough to fit through the holes. It will keep moving in a straight line through the holes due to its tangential velocity. If there were no holes, then the water would have a centripetal force and stay inside the washer.

Center of Gravity 

Example: Wrestlers bend their legs when wrestling, because it lowers their center of gravity. This makes it harder to get their center of gravity outside of their base of support, thus making it harder to create a torque. It essentially makes them more stable.

Example: A handy-man has wrench on a stuck bolt and can't seem to get the bolt loose. How can he make his job easier? He should get a longer wrench, because that will create a longer lever arm (the distance from the force applied to the axis of rotation). Now there is a larger lever arm AND a larger force, so there is a larger torque. Torque = (F)(lever arm)

Example: If a steel ball and a tape roll were to race down a ramp, the steel ball would win. The steel ball's weight is evenly distributed throughout the object, while the tape roll has a hole in the center so its mass is farther from its axis of rotation, giving it a higher rotational inertia and thus a smaller rotational velocity. The steel ball will have a smaller rotational inertia and thus a larger rotational velocity.

Example: If a water bottle filled with frozen water and a water bottle filled with regular water were to race down a ramp, the regular water bottle would win. In the frozen water bottle, everything is rotating when the bottle rotates, because it is all frozen together. In the regular water bottle, the bottle is rotating around the water, because the water for the most part stays in place.