Rotational KE leads to moment of inertia

• The kinetic energy of a rotating body depends on its angular velocity (squared) and its moment of inertia.
• Moment of inertia is a simple sum if a body has discrete sub-units.
• Moment of inertia is an integral if a body has continuous segments.
• If you know the moment of inertia around an axis through the center of mass, then you can find the moment of inertia around any axis parallel to the first via the parallel-axis theorem.

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  Let's try an easy calculation: what's the moment of inertia of these three balls? Each ball has mass m = 3 kg, and they are arranged in an equilateral triangle with sides of length L = 10 m. We spin the triangle around the spot marked "X", which is one of the balls.

  

  Okay, let's try a harder one. Suppose that we locate the center of mass of the triangle -- I've marked it with the "X" in the figure below. What's the moment of inertia of the triangle spun around this point?

  

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