PX155 - E14 - parallel axis theorem

moment of inertia of an object of mass $m$ around an axis that lies at a perpendicular distance $d$ from the object's centre of mass equals the moment of inertia around a parallel axis through the centre of mass plus $m d^{2}$ :

I (a n y a x i s) = I_{c m} (| |) + m d^{2}

avoids needing to recompute $I$
eg: $I_{c m}$ for a long thin rod mass, $m$ , length about an axis through the centre perpendicular to rod:

I_{c m} = \frac{1}{12} m L^{2}

- what is $I$ for rotation about a parallel through one end?
- here, $d = \frac{L}{2}$ :

I = \frac{1}{12} m L^{2} + \frac{1}{4} m L^{2} = \frac{1}{3} m L^{2}

justification of the theorem

I_{A} = \int (r^{'})^{2} . d m

cosine rule:

(r^{'})^{2} = r^{2} + d^{2} - 2 d r \cos θ

I_{A} = \int r^{2} . d m + d^{2} \int d m - 2 d \int x . d m

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