From parallel axis theorem follows that, given a rigid body, the moment of inertia is minimum if calculated with respect to an axis passing through the center of mass. What is the physical meaning of this?
The moment of inertia plays the role of the mass on rotational motion, so it is linked to the reluctance of the body to rotate. Given a torque, if the axis passes through the CM the angular acceleration is maximum, while it decreases increasing the distance from CM. Why does that happen?
I tried to justify it as a consequence of the distribution of mass around the CM, but I do not know if that is right.
Best Answer
There are a few ways to justify it.
First, you could look at the motion of the object as it rotates. In 2D, it turns out that all such motion can be decomposed into motion of the CM, and rotation about the CM. Therefore, rotating around the CM itself is the only way to guarantee the CM doesn't move, and hence is the least energetically costly.
Another way is by direct calculus. Suppose some 1D object is made of point masses at positions $x_i$ with masses $m_i$. Then $$I(x) = \sum m_i (x_i-x)^2$$ is the moment of inertia about $x$. The minimum is attained when the derivative is zero, so $$0 = 2 \sum m_i (x_i - x)$$ which implies that $$x = \frac{\sum m_i x_i}{\sum m_i}.$$ This is the definition of the center of mass.