**Rotation**

To this point we have used a particle model

for our study, for the most part at least.

This chapter branches out a bit and we

start to study rigid bodies, and rotation.

A rigid body is one that doesn't bend,

think of things like a disk, a ball, a gear

merry-go-rounds.

We will also play with divers, ice skaters,

and other such people in motion.

Now they aren't strictly rigid, but for

some of their motion we'll pretend.

Rigid bodies have two basic types of motion.

Translational: We have used this to great extent.

Blocks slide down ramps, projectile motion, etc

Rotational motion: we also did this to some extent.

Disks spinning, objects turning.

In addition to these two types a combination can be made

as well.

This hammer has a translational motion

as well as a rotational motion.

maybe easier to see is this wrench. It is sliding across a desk

in a straight line, but it is also rotating.

For a bit of review we have rotational quantities

and their relationships

With sign convention as well

Further review, we have rotational kinematic equations

Finally the connection between rotational

and translational motion.

A high speed drill reaches 2000 rpm in 0.5 s

What is the drill's angular acceleration?

Through how many revolutions does it turn during this first 0.5 s?

As we begin to talk about rigid body

rotation, we need to talk about what

the body rotates around.

this is the center of mass.

You can see this in the previous pictures.

Here is another example that illustrates this motion.

What then is the center of mass?

Be careful it is not just the center of the object

it is the mass weighted center.

Meaning that you find where all the mass

of the object is and then find the center of that.

So, how do you do that?

How can you find the center of mass of an object?

Here are three methods

1. Spin the object.

It will rotate around the center of mass.

think back to the youtube video.

2. Hang the object by 2 or 3 different

locations, and draw a line straight down

from the fixed point.

The intersection of the lines will be the

center of mass.

3. Use an equation

This equation assumes that you can add up every single

particle and its mass at its location. in practicality , not good.

Your book then goes into a bit of calculus and

takes the limit of this summation to an integral

and gives you the integral form of the sum.

This is still not a practical equation.

You would need to know an equation describing

the mass density of an object. As well as a perfect

size or dimension of an object.

This would work great in uniform rods, and circles,

but for any engineering purpose, no good.

Also note that in the current form this integral

is not helpful, you need a dx or a dy.

For a uniform rod you could consider

that the ratio of mass and distance over the total

mass and distance are the same.

This would give:

The three masses shown are connected by massless rigid rods. What are the coordinates of the center of mass?

**Questions from your studying**

If I throw a hammer, or tennis racket, and asked you to find the velocity of any point on it at any time,

could you do it?

An 18 cm long bicycle crank arm, with a pedal at one end is attached to a 20 cm diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s.

A. What is the tangential acceleration of the pedal?

B. What length of chain passes over the top of the sprocket during this interval?

419 rad/s^2

8.3 rev

.057 m/s^2

7.9 m

100 g

200 g

300 g

10 cm

10 cm

**A 100 g ball and a 200 g ball are connected by a 30 cm long massless rigid rod. The balls rotate about their center of mass at 120 rpm. What is the translational speed of the 100 g ball?**

(6.67, 5.0)

**2.5 m/s**

"Why don't radians have units attached to them? And why haven't we created a unit that could be attached to it?"

"How are radial acceleration and tangential velocity related? Why are Radial acceleration and radial velocity not related?"

"Why is it that rotational motion is not rotation about the center of mass?"

"Can we go over rotation about the center of mass and equation 12.4?"