Parallel and perpendicular axis theorems
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Parallel Axis Theorem
We have obtained standard results for the moment of inertia for certain bodies about an axis passing through each body's centre of mass. We now need to find the moment of inertia of a body about an axis at a distance d from the first axis. Rather than working from first principals every time we can employ the parallel axis theorem. We proceed to state this theorem, illustrate its use and then prove it. Parallel Axis Theorem Suppose the moment of inertia of a body M about an axis passing through its centre of mass is Mk2, then its moment of inertia about an axis parallel to this first axis but at a distance d from it is[Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Example (1) A rectangular grid is made of four thin uniform rods. The rectangle has length 4a and width 3a. The longer pieces have mass 4m and the shorter pieces have mass 3m. Find the moment of inertia of the grid about an axis through one of its corners and perpendicular to its plane. [Diagram goes here - download the original to see it.] Label the four pieces A, B, C, D with centres of mass XA, XB, XC, XD respectively. Then MA = 4m, MB = 3m, MC = 4m, MD = 3m. About XA, XB, XC, XD moments of inertia are [Equation goes here - download the original to see it.] By the parallel axis theorem [Equation goes here - download the original to see it.] Therefore [Equation goes here - download the original to see it.] We now proceed to prove the parallel axis theorem.
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Contents of
Parallel and perpendicular axis theorems
1 Parallel Axis Theorem
2 Proof of the parallel axis theorem
3 Perpendicular Axis Theorem
4 Proof of the Perpendicular Axis Theorem
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