UNIT 4 OVERVIEW

In Unit 4, we covered 6 major topics:

1. Rotational and Tangential Velocity
2. Rotational Inertia
3. Conservation of Angular Momentum
4. Torque
5. Center of Mass/ Gravity
6. Centripetal/Centrifugal Forces

Rotational and Tangential Velocity

We measure linear motion in speed. For circular motion, however, we have to ways to express velocity:

• Tangential Velocity
• Rotational Velocity

Rotational Speed measures the number of rotations that occur in a certain unit of time.  For instance, A car measures a it's speed in "RPM", or Rotations Per Minute.

Tangential Velocity measures the distance covered per unit of time such as miles per hour (mph) kilometers per hour (km/h).  Therefore, it is still linear speed, but in a circular path.

One thing to remember about tangential velocity is that it depends on RADIAL DISTANCE.  Radial distance is distance from the AXIS OF ROTATION.  They are directly related.  Also, Tangential Speed is also directly related to rotational velocity.  We express both these relationships like so:

v ~ r w

(tangential velocity ~ radial distance * rotational velocity)

-We saw this relationship when we did the activity outside.  We linked arms and created a line, rotating around Maiya as our axis of rotation.  As people moved from closer inside of the circle to the very outside of the circle, they noticed they must walk much faster, even run, to keep the same number of rotations as the people in the inner circle.  

-For instance, the image below shows 4 students on a ferris wheel, Maiya, Catherine, Princess, and Jasmin climbing the bars

Jasmin and Princess will experience the same number of rotations, however, Princess will have to go faster (have a greater tangential speed) to reach them.  This is because her radial distance is greater.

Rotational Inertia
Rotational inertia is a property of an object to RESIST change in spin.
Rotational inertia is similar to inertia, except rather than depending on mass, it depends on DISTRIBUTION OF MASS.

The two dumb-bells shown above have equal weights, however the let's weight is distributed father from the axis of rotation than the right's.  Therefore, it has a larger rotational inertia, and is harder to spin.  The right dumb bell has a smaller rotational inertia, therefore is easier to spin.

Here's a video we created on rotational inertia:

Conservation of Angular Momentum

when calculating angular momentum, we use:

• mv before = mv after

Angular momentum is similar, but now we will use rotational inertia (RI) and rotational velocity (RV).  (these are not the real letters we use, but we will use them now for sake of space)

• RIbefore * RVbefore = RIafter * RVafter

This image shows a spinning ice skater.  We know their momentum before must equal their momentum after (conservation).  As they take their arms in, they are decreasing their rotational inertia, which will increase their rotational velocity.

Torque

Torque is the force that causes rotation.  

Torque = force * lever arm

Lever arm is the distance from the axis of rotation.

This is why we use wrenches of different lengths.  The longer wrenches have a greater lever arm, therefore require less force to unscrew a bolt.

Torque also affects balance which will be explained in the section below...

Center of Mass/ Gravity

The center of gravity for any object is it's average position of all mass

The base of support is the plane on which the object is supported by the ground.

An object will fall if it's center of gravity does not align over it's base of support:

There are two ways you can make something more stable:

1. Create a bigger base of support
2. Lower the center of gravity

This is why wrestlers bend spread their legs (#1) and bend their knees (#2), so they can't be pushed over as easy.

Centripetal and Centrifugal Forces

A centripetal force is a center- seeking force that keeps something moving in a curve:

Why don't you continue forward when your car turns?  Because of Newton's 1st law, your body wants you to.  However, the car door runs into you (centripetal force) and you run into the door (centrifugal force).  When turning, it may feel like you are the one pushing on the door, but this is not a real force.  It is named centrifugal force, however it doesn't really do anything.

Meter Stick Experiment

In this lab, our goal was to find the mass of a meter stick using only the meter stick and a 100g lead weight.

The purple dot on the ruler shown above is the center of gravity (or mass).  The ruler above is balanced on the table because it's center of gravity is directly above the base of support (the table).  Therefore, there is no lever arm and no torque to make the ruler rotate at all.

This illustration shows a ruler with a lever arm.  In reality, this ruler would rotate counterclockwise.  Since the ruler's center of gravity is not over a base of support, there is a lever arm (shown in green).

As we know, torque depends on two factors:

Torque = Lever arm * Force

In class, we added a force to the left end of the ruler by adding a 100g weight. 

For the ruler to be balanced, the center of gravity of the ruler must be moved back to increase the lever arm.

Force * Lever arm   =  Force * Lever Arm

TO FIND METER STICK'S MASS

-First, you must find the weight of the 100g weight because that is the force.  You use the w=mg equation, using 9.8 as g and .1 as the mass (because it is 100g but in kilograms).  Therefore, the force from the weight is .98 N.

-Next, to find the lever arms of both sides, you measure the distance from the weight to the table:

-Then, we found the other lever arm by measuring from the point on the table to the center of mass of the ruler:

We measured both these lengths and got these results:

                Lever Arm 1: 24.5

                Lever Arm 2: 25.5

We then entered these measurements into the toque equation like so:

Torquecounter clockwise = Torqueclockwise

Force * Lever Arm = Force * Lever Arm

.98 * LA1 = w * LA2

(.98)(24.5) = w(25.5)

w = .94 N

To then convert this to grams, we plug it back into the w=mg equation, and get .096 kg, which converts to 96 grams.

We then weighed the meter stick and got 95.7 grams

Therefore, we were only .3 grams off!

Here's how to explain this lab on a quiz or test:

Center of Gravity Experiments

This video shows 3 different experiments finding and testing center of gravity.  Although it's hard to hear the narrator, this is still an interesting video of how we can see center of gravity in real life.

Angular/Rotational Momentum Video

This video provides a good explanation of what angular momentum is, how the law is used, and how it can be applied to everyday life using explanatory animations and video clips.