Moments of Inertia (Oct. 25, 2010)

Moment of inertia, also called mass moment of inertia,or rotational inertia, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. Moment of Inertia in SI units is kg·m².

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. The moment of inertia depends upon how an object's mass is distributed relative to its pivot point.

For a point mass the moment of inertia is just the mass times the square of perpendicular distance to the rotation axis, I = mr². That point mass relationship becomes the basis for all other moments of inertia since any object can be built up from a collection of point masses.

For an object with an axis of symmetry the moment of inertia is some fraction of that which it would have if all the masswere at the radius r. For a continuous mass distributions it requires an infinite sum of all the mass moments which make up the whole thing, is the integration of all the mass.
In class we discussed that the dm can be expressed in many other forms for example:
dm= pdv
dm=(lambda)dL
dm=(m/L) dL
dm=(m/L) dr
dm=(lambda)dr

(lambda= linear density or the mass per unit length)

The moment inertia about the end of the rod can be calculated directly or obtained from the center of mass expression by using the Parallel Axis Theorem.
The moment of inertia about any axis parallel to that axis through the center of mass is called the Parallel Axis Theorem given by:
*Note if you know the moment of inertia of the center of mass you can find the moment of inertia about any pivot point using the parallel axis theorem, given that the axis of rotation is parallel.

Six steps to find Inertia....
1. Draw a diagram & choose the point of rotation.
2. Write the definition of moment of inertia.
3. Use density to find dm as a function of dr.
4. Plug in dm as a function of dr into the integral.
5. Choose appropriate bounds.
6. Integrate.

We also compared in class the similarities of liner and rotational in respect to inertia ...
ere is a brief summary diagram:





Here is an example of the derivation of a thin disk with x,y, and z axis.


*resources : class notes
http://hyperphysics.phy-astr.gsu.edu/HBASE/mi2.html