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24.3 MOMENTS OF INERTIA
À pWhen we considered the centroids/centre of mass we assumed that the force acting on each chunk was the same (uniform gravity).
If we have a force that varies over a surface linearly we use the second moment of inertia.
Consider a simple case of a beam (not quite accurate). The material in the beam acts as small springs. As the beam is bent the material in the centre is almost neutral, but towards the sides it is compressed or stretched.
If we consider the beam above as a continuous force, instead of distributed springs it changes the solution (by analogy you might recognize that with centroids we could lump forces, but with second moments we cannot).
Another example of a linearly varying force is hydrostatic pressure based on water depth. Basically we add the mass of the water directly above to find the pressure on an area.
The general forms of moments resulting from linearly varying forces are,
We generally calculate second moments of inertia using,
as an example we can consider a triangle. The calculations will be to find the second moments of inertia about the x and y axis - not the centroid.
The polar moment of inertia may be simple to calculate with rotational parts, but it can also be found using a simple relationship (note: the polar `J' and the `o' for the origin)
After we have found the second moment of inertia about an axis, we can find it about another parallel axis using the parallel axis theorem.
Consider the application of the parallel axis theorem to the triangle seen before, To find the moment of inertia about the y centroid, when all we have is the y moment of inertia about the x axis.
NOTE: when using the parallel axis theorem, the centroid should always be used as a reference.
Like centroids, we can calculate moments of inertia for simple areas using weighted sums (a slightly different technique to finding centroids). The basic steps are,
1. For each simple shape find the moment of inertia about some global axis. This may require the use of centroids and areas to move the axis.
As an example, let's consider a problem (9.26 pg. 64 Beer and Johnston)
24.3.1 Practice Problems
(|1. Find the Second Moment of Inertia about the centroid for the beam section shown below.
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