Moments of Inertia

Moments of inertia for planar symmetric objects

In this section, we find moments of inertia for an important class of objects: symmetric planar regions. The symmetry enables us to compute moments by solving a single integral.

The most important fact about moments of inertia is this:

The moment of inertia of an object depends on the axis of rotation.

A very important moment of inertia is called the centroidal moment. This quantity tells us how hard it is to rotate an object about an axis passing through the object's centroid. Typically, engineers compute moments of inertia geometrically, meaning that they are interested in the moments for regions with constant density p = 1. In this section we will follow this approach. Thus, to find the moment of inertia about the centroid for a symmetric region

we integrate

where y1 is the distance from the centroid to the bottom of the object and y2 is the distance from the centroid to the top of the object.

Question 1

1. Find the centroidal moments of inertia for
   1. a rectangular beam of total width w and height h
   2. a T-beam as in
      
   3. a C-beam as in
      
2. Can you use properties of integrals to derive a simple relation between the centroidal moment of inertia for a composite region and the centroidal moment of inertia for the component pieces? You may assume that you also know the vertical distances between the centroids of the components and the centroids of the composite regions.
   If you think a relationship exists, state it and test your conjecture on the T-beam and C-beam cross sections.
   If you think no relationships exists, determine the mathematical features of the problem that prevent a simple relationship (eg, nonlinearity).

Question 2

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