Moment of Inertia
The moment of inertia (also called the second moment) is a physical quantity which measures the rotational inertia of an object.
The moment of inertia can be thought as the rotational analogue of mass in the linear motion.
The moment of inertia of a body is always defined about a rotation axis.
Moment of Inertia of Point Masses
For a single mass, the moment of inertia is expressed as
where
If a system consists of
where
We can represent the last equation in the form
where
It follows from the above equation that
When all masses
then the moment of inertia can be written in the form
In this case,
where
Moment of Inertia of a Lamina
When we deal with distributed objects like a lamina, or a solid, we need to calculate the contribution of each infinitesimally small piece of mass
Case 1. Density Depends on the Coordinate
Let a planar lamina be bounded by the curves
If the density
The total moment of inertia of the lamina about the
Case 2. Density Depends on the Coordinate
Similarly, we can consider a region of type
Parallel Axis Theorem
Suppose that an object is rotated about an axis passing through the center of gravity of the object and has the moment of inertia
where
By definition, the distance
Solved Problems
Example 1.
Find the moment of inertia of a rectangle with sides
Solution.
Consider a small strip of the rectangle of width
Assuming the density is
Integrating from
Example 2.
Find the moment of inertia of the semi-circular arc of radius
Solution.
We take the radius vector that forms an angle
The moment of inertia of the element
Recall that
To calculate the total moment of inertia of the semi-circular arc, we integrate from
Example 3.
Find the moment of inertia of a uniform thin disk of radius
Solution.
Take an arbitrary thin ring of radius
where
The moment of inertia of the ring is given by
To find the moment of inertia of the entire disk, we integrate from
Note that the mass of the disk is
so
Example 4.
A thin uniform rod of length
Solution.
First we determine the moment of inertia of the rod about the axis passing through the center of gravity.
Consider a small portion of the rod located at distance
where
The total moment of inertia is defined through integration:
Since the mass of the rod is
we have
Suppose now that the rod is rotated about the axis passing through one of the ends.
The new axis is located at a distance