ROTATIONS AND THE MOMENT OF INERTIA TENSOR 1. Rotation ...

ROTATIONS AND THE MOMENT OF INERTIA TENSOR
JACOB HUDIS
Abstract. This is my talk for second year physics seminar taught by Dr.
Henry. In it I will explain the moment of inertia tenssor. I explain what the
tensor is used for and how it is used. I will also explain both physically and
mathematically how and why an object can precess.
1. Rotation of two equal masses connected by a spring (about the
C.M. of the system)
Here are two equal masse connected by an unstretched sprong.
If both masses are given an impulse (kick) of equal magnitude but opposite direction
(one into the page and the other out of the page) the spring will elongate
and the masses will start to rotate in a circle about the center of mass of the system.
Both masses have a force pulling on them. The force keeps both of the masses
moving in univorm circular motion.
F 1 = m1(ω 2 )r1
F 2 = m2(ω 2 )r2
Date: Wednesday, April 05, 2005.
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It is important to note the the angular momentum L and the angular velocity ω
are in the same direction. (They point up to the top of the screen.)
If this system of two masses connected by a massless spring was put in space
(far from external forces) and given the correct initial conditions, it would rotate
in uniform circular motion forever (unless acted on by and external torque).
2. Rotation of a thin plate
Imagine a plate is rotated about its lower edge with omege (fix) in the y-direction.

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One can ask the question ”What is the angular momentum of the plate about its
lower left hand corner?” (you can ask that question about any point on the plate,
but I will calculate it for the angular momentum about the lower left hand corner.
Remember, L = rxp where r is a vector from one point in space.) The way you
calculate the angular momentum of an extended object is to brake the object (plate
in our case) up into many small area (or mass) elements and sum the angular momentum
of each one. This gives the angular moment of the entire object about a
given point.
For the plate
L = ∑ n
i=0 r ixp i
In the above equation, x is the cross product. At this instance in time (the previous
picture), p i = mv i = mz i ω in the x direction. (You have to imagine braking
the plate up into N small squares and labeling each one 1 to N where i is the index
of the square you are talking about).
And equation for angular momentum turns out to be
n∑
L = (y i ŷ + z i ẑ)x(mz i ωˆx)
i=0
n∑
[m i ω(z i x i ŷ − y i x i ẑ)]
i=0
You can see from the above equation that L (angular momentum) and ω (angular
velocity) are not in the same direction.

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If a plate is given the initial conditions that it is being spun about its lower edge
with /omega in the /haty direction, the book will not continue spinning that way.
Someone has to put an external torque on the book to keep it spinning that way.
You can see this because L is not constant in time. As the plate spins (assuming it
keeps spinning with /omega in the y-direction /omega will form a cone about the
y-axis. The net torque is equal to the time derivative of the angular momentum and
because the angular momentum would be changing in time, someone or something
would have to exert and external torque on the book for it to keep spinning that way.
To sum it up, if a book is given the initial conditions such that it is being spun
about its lower edge with ω in the ŷ direction, the book would not continue spinning
that way; something else would happen.
3. The inertia tensor
The inertia tensor allows one to calculate the angular momentum of an object
when that object is spun about any axis (i.e. about any point). To use the inertia
tensor, you must specify the orientation of a coordinate system (thus it is a tensor).
The form of the inertia tensor is the volume integral of the below matrix times
the density (look it up in any mechanics book).
⎡
⎤
⎣ (x2 + y 2 ) −xy −xz
−yx (x 2 + z 2 ) −yz ⎦
−zx −zy (x 2 + y 2 )

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3.1. an example. Here I am going to calculate the moment of inertia tensor for a
plate. I will pick a yx-cartesian system with the origin at the left hand side of the
plate.
I will show how to calculate the I zz component of the above matrix and the
others are done the same way.
I zz = ∫ (x 2 + y 2 )dv this is = 2a 2 m/3
The inter inertia tensor is as follows.
⎡
⎣
1/3 −1/4 0
−1/4 1/3 0
0 0 2/3
Notice that the inertia tensor (for this example) is not diagonal. This means that
if you rotate the plate along x or y, L and omega won’t be in the same direction.
For the given cartesian system, x and y are not principle axis.
another way to see this is if you multiply the above matrix by the vector
and you will get
⎡
⎣
⎡
⎣ wx
⎤
0 ⎦
0
1/3wx
−1/4wx
0
This demonstrates mathematically that L and W are not in the same direction.
It is possable to diagonalize the inertia tensor and find the principle axis. If you
do diagonalize the above matrix (you should try it out) one of the eigenvalues you
will get (moment of inertia value) is I=1/12. The eigenvalue corresponding to the
moment of inertia you get is:
⎤
⎦
⎤
⎦

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⎡
⎣ frac12.5
−1/4wx
0
⎤
⎦
If the plate is rotated by ω = frac12(x+y) L and W will be in the same direction
and I (the moment of inertia) will equal 1/12.
The picture of this is given below.

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Every point on the plate is kept in circular motion by a force pulling it to the
center. The net force on the plate is zero because there is an equal and opposite
force balencing each particle on the plate. This is because there is another particle
on the oppside of the plate (mirror symetry) that is also moving in univorm circle
motion. If the plate is set up such that it is spinning along its diagonal (or any
of its principle axis; the diagonal is not the only one) it would rotate, in uniform
circular motion, forever, unless acted on my an external torque.
3.2. precession. The idea of precession is simply this: IF AN OBJECT IS NOT
ROTATING ABOUT ONE OF ITS PRINCIPLE AXIS ITS ANGULAR VELOC-
ITY WILL BE CHANGING WITH TIME EVEN THOUGH NO TORQUE WILL
BE APPLIED TO THE OBJECT. This seems to violate the formula T = Iω However,
that is not correct because this formula only works if I is a principle moment
of inertia and the torque is parallel to the principle axis. Below I give an example
to illestrate this idea.

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A massless rod connects 2 equal masses. THE COORDINATE SYSTEM WE
ARE WORKING IN IS Z-UP/DOWN, Y RIGHT/LEFT AND X INTO AND OUT
OF THE PAGE.
Imagine the
ω = w xˆx + w z ẑ
(obviously w y = 0)
L (angular momentum can always be written in the following form
L = I 1 w 1 + I 2 w 2 + I 3 w 3
In this formula I i where i can be 1, 2 or 3 are the moments of inertia of an
object about a STATIONARY REFERENCE FRAME (CARTESIAN SYSTEM).
It is easiest to use 3 principle moments of inertia.
So, for the given example
L = I 1 w 1 + I 3 w 3
You can see from the picture that
I 1 = 2mr 2 (the rod is 2r in length) and I 3 = e where e is a very small number
This is the moment of inertia of the rod at time t=0 seconds. A small time
later, the rod has rotated a little about x and a little about z and it has changed
positions. It will look something like this:

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This picture is exagerated so you can see what is happening (In real life after a
short time the angle wouldn’t be anywhere as near as large as I have drawn it to
be.). This object has been given an angular velocity
W = w x hatx + w z hatz
in a location far from any external influences. The net torque acting on this object
is zero therefore the angular momentum L must be constant. As I stated before,
the angular moment of this object with the above given angular velocity is
L = I 1 w 1 + I 3 w 3
At time t=0 seconds I 1 = 2mr 2 and I 3 = e Now, look at the above figure. Later
in time, I 1 = 2mr 2 , it has not changed but what about I 3 ? You can see from the
picture that I 3 has increased. The perpendicular distence from the z-axis to one of

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the masses has increased. ( I 3 >> e)
There is no torque acting on the two masses. So, angular momentum L has
to be constant in time. But, as I have just shown the moment of inertia of the
object(about its center of mass) is changing with time. The moment of inertia is
given by the formula
L = I 1 w 1 + I 3 w 3
So, even though the net external torque on the system is =0, the angular momentum
of the system is not a constant but rather changes with time.
The main poing: If an object is not rotated about a principle axis, its angular
velocity will not remain constant in time even though its angular momentum and
torque are.