In this video we will introduce mass moment of inertia and area moment of inertia. 
The calculation of the area moment of inertia will be shown in some examples in the next video. 
Some students find these concepts very abstract and difficult understand.
By introducing them together I hope it will help you to understand these concepts.
But before that I want you to pause for a moment and think:
what does the word inertia mean?
In what capacity have you already known about inertia?
When you think about it what physical quantity comes to mind?
[PAUSE PLEASE]
The answer is, inertia is the resistance of an object to changes in its state of motion,
and the physical quantity that you are already familiar with is mass. 
But what causes motion?
We know that force causes translational motion while moment causes rotational motion. 
According to Newton’s second law, force equals to mass m times linear acceleration. 
Or it can be written this way, that the acceleration a is caused by force F, and it equals to F divided by the mass of the object m. 
As you can see, here mass is the resistance to translational motion. 
And that’s why sometimes mass is also called inertia mass. 
And since moment causes rotation, 
can we write something similar and relate moment to the rotational motion of an object? 
Indeed we can.
Similar to Newton’s second law, moment equals to I_m times the angular acceleration alpha.
As you can see, moment and angular acceleration are both vectors and they have the same direction. 
Again, we can rewrite this equation into this form. And if you compare it to the equation above,  
you can tell that this term I_m is the resistance to rotation, 
and it is known as the mass moment of inertia. 
Note that I put the subscript m here because I will also discuss area moment of inertia in this same video and I want you to be able 
to distinguish between them. But often times you will see it without the subscript m. 
How is it defined? 
Just like moment caused by a force, the moment of inertia is always calculated about an axis, say, axis a-a. 
On the rigid body, for an arbitrary differential element with the mass dm, 
we can find the perpendicular distance between it and the axis, r, 
and the mass moment of inertia about this particular axis a-a is the integration of r squared dm, integrated throughout the entire body. 
So unlike mass, which is absolute,
mass moment of inertia is relative, and it’s different when calculated about different axis. 
Needless to say, in this moment equation, these two must be calculated about the same axis. 
So that’s the mass moment of inertia. 
What about the area moment of inertia?
We know that for an object with certain mass, 
if it has uniform density rou, then mass equals to rou times the volume V, and we can use a volume to represent the object.
And if the volume has one uniform dimension, say uniform height h, then V is h times the cross sectional area A,
and we can use the area to represent the object. 
The area can be considered as a geometric reduction of the object. 
Similarly, if the object has a mass moment of inertia calculated about a specified axis,
because dm equals to density rou times the differential volume dV, 
then this can be written as
constant rou times the integration of r squared dV throughout the volume of the object. 
And again, for constant height h, we can pull this constant outside the integration sign, and get this, which is the area moment of inertia. 
So, mass can be reduced to a 2-D geometric representation, the area,
and similarly, mass moment of inertia can be reduced to a 2-D geometric representation,
the area moment of inertia.
I will give examples in the next two videos on how to calculate the area moment of inertia. 
But before that 
I want to remind you that a lot of concepts are easier to understand if you consider them as counterparts. 
The first row of this table deals with translational motion and the second row deals with rotational motion. 
For the driving force, we have force versus the moment of the force.
For displacement we have linear displacement s versus angular displacement, theta, which is an angle.
For velocity we have linear velocity v versus angular velocity omega.
For acceleration 
we have linear acceleration a versus angular acceleration alpha. 
For inertia, or resistance to motion, we have mass versus mass moment of inertia. 
And lastly, for the 2D geometric reduction of inertia we have area versus the area moment of inertia. 
Hopefully this table can help you better understand the concepts of the moments of inertia. 
