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In this problem, you're asked to find the ratio of frequencies about two pendula: one with 2M at the end of a massless rod, the other with 1M at the end and 1M in the middle of the rod. I tried to do this problem by locating the CM in the second case and treating it as a pendulum of shorter length. This fails to work though. When you work out the problem with sum of torques and calculate the moment of inertia as two point masses, it works. Why does the CM not work? I get a CM of (3/4)*r, and I get a moment of inertia of 2M*(3r/4)^2 = (9M/8)r^2
Okay, so you can't calculate a moment of inertia via the center of mass. I just remembered this, but why can't you do this? Young and Freedman, pg 341 reminds you of this, but it doesn't tell you why it fails, it only gives you an example of its failure. Whats the deal here?
The reason you can't just use the cm is because the moment of inertia is proportional to r^2, not r. If it were linear in r, then you could use cm. Just like how kinetic energy is proportional to velocity squared, you can't just take the average velocity of a system and use that to calculate the total kinetic energy. If you did, you would get zero kinetic energy for a gas, for example. Now, imagine two balls connected with a rod rotating about the center of mass. Using center of mass, you would get zero angular momentum.