[Physics] Moment of inertia of disc with a hole

Question

Suppose we have a disc with a hole, when computing moment of inertia of this about the disc's centre. Why do we subtract the moment of inertia of the removed part from the moment of inertia of complete disc?

Answer 1

Because the moment of inertia for a point mass is:

$$ I = mr^2 $$

When calculating the moment of inertia for continuous bodies we use calculus to build them up from infinitesimal mass elements, so effectively to calculate the moment of inertia of the disk (without hole) we're doing:

$$ I_{disk} = \sum_i^{disk} m_ir^2 $$

for the collection of infinitesimal masses $m_i$ that make up the disk. When you create a hole you simply subtract off all the point masses in the hole to remove them from the sum:

$$ I_{net} = \sum_i^{disk} m_ir^2 - \sum_j^{hole} m_j r^2 $$

where the second sum is over all the infinitesimal masses in the bit you're removing to make the hole. So you end up with:

$$ I_{net} = I_{disk} - I_{hole} $$

PS as Stefan points out in his comment, the $I_{hole}$ above is the moment of inertia about the axis of the disk not about the axis through the centre of the hole.