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# PH 121 12.3-12.4

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## Richard Datwyler

on 7 July 2015

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#### Transcript of PH 121 12.3-12.4

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

"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
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