**Rotational Energy and Mass**

In speaking of energy, there is both energy of position and energy of

motion.

When an object rotates it moves. Amazing.

Thus it has kinetic (moving) energy.

Yet if it is rotating the 'translational or tangential' velocity

is related to angular velocity.

With these two together we have a

rotational kinetic energy.

Note this is specifically of only one mass

at one position.

this would be like a rock in a sling

However, most objects have mass and

distances that vary.

Thus you would have to add them all up.

This summation is an important term that

is specific for different types of objects.

It is called the moment of inertia.

Thus we can write the rotational

kinetic energy as:

This moment of inertia is analogous to

to the linear inertia that we have had before.

They both depend on the mass of the object

but when you rotate it also depends on how

far away that mass is from the axis of rotation.

Consider trying to rotate a 2x4 wooden board

by the end of the board, versus

by the middle of the board.

It has the same mass, but the mass is further away

from the axis of rotation, thus it has a larger moment of

inertia.

Demo

We could have much fun

with integration, finding the moments

of inertial for different objects.

Here is a list from your text.

With these comes the ability to also

find moments of inertia that are

not in the center of an object, or at

the edge.

This leads to the parallel-Axis theorem

To find the moment of inertia of any object we

need to change our sum into an integral

In doing this we need to somehow compare

the r and the dm.

for example consider a disk.

As I go out radially the mass scales the same as the

area scales.

Here the area of the disk is

and a little small area can be

thought of by unwinding some

small ring having a height and length

multiplying them together gives:

Thus

Back to where we started :

Combining gives

Doing the integral gives:

Which is the same as the table.

This process can be done any number of

times for any number of objects

I will not test you on this.

It is an intermediate level skill.

Yours is to learn the concept of moment

of inertia, and use a table to solve problems.

The parallel - axis theorem

says that the moment of inertial

about an axis that is parallel to the

center of mass can be solved as:

Where M is the mass of the object

and d is the distance from this

parallel axis to the center of mass

And that is where we will leave off for today,

any last questions for me before the majority

of you take the test?

**Which has**

a larger moment of

inertia

Solid sphere or

Solid Cylinder

A sphere

B Cylinder

C need more info

a larger moment of

inertia

Solid sphere or

Solid Cylinder

A sphere

B Cylinder

C need more info

"Do we have to memorize the Moment of Inertia formulas for all the different types of objects?"

"Could you go over how to find the mechanical energy?"

"Can you explain equation 12.14 ? (energy)"

"why is kinetic energy not dependent on gravity?"

"Why are the moments of inertia for hollow circular objects greater than the more massive solid objects?"

"Stop and think 12.1"

A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door's moment of inertia for

a. Rotations on its hinges and

b. rotation about a vertical axis inside the door, 15 cm from one edge?

6.9 kg m^2

4.05 kg m^2

A 1.5 m rod with a mass of 300 g is able to hinge on one end. It is initially horizontal, but is able to pivot until the rod is vertical, stopping as it hits a barrier. How fast is the tip of the rod moving when it hits the barrier?

6.64 m/s