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QUESTION

# How do you calculate the moment of inertia of a beam?

Basically, we need to add up all of the products of ##m*r^2##.

This depends on a few factors, but here we will look at the most common two types of problems. If you would like more complex problems - ask for beams without uniform mass distributions.

The formula for the is ##I=summ*r^2##, but there is a problem with this because a beam does not simply have a whole bunch of point masses. Rather, the mass is distributed throughout the whole beam. You will address this in more detail in calculus, but let's look at uniform beams.

Most common case 1: Find the moment of inertia of a uniform beam of 4 kg that is 2m long about its CENTER.

Most common case 2: Find the moment of inertia of a uniform beam of 4 kg that is 2m long about its END.

How most physics students need to proceed is by using a table of moments of inertia, such as these: http://en.wikipedia.org/wiki/List_of_moments_of_inertia.

You see here, they are calling the first case "Rod of length L and mass m", and they have ##I_(center)=(mr^2)/(12)##, so we would get

##I_(center)=(4kg*(2m)^2)/(12) = 1.33kg*m^2##

For the second case, we have ##I_(end)=(mr^2)/(3)##, so we would get

##I_(center)=(4kg*(2m)^2)/(3) = 5.33kg*m^2##

You can clearly see that WHERE the rotation occurs can make a big difference on the moment of inertia.