Parallel and perpendicular axis theorems
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Proof of the parallel axis theorem
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Let p be a body (rod, lamina or solid) of mass M with moment of inertia I = Mk2 about a perpendicular axis z passing through its centre of mass. Divide p into small segments each of mass M. Let ri be the distance of the mass segment Mi from C, the centre of mass of p. [Diagram goes here - download the original to see it.] Consider the weight Wi of the mass segment Mi. The torque produced by this weight about the axis z is [Equation goes here - download the original to see it.] [Diagram goes here - download the original to see it.] Here [Equation goes here - download the original to see it.] is the angle made between the perpendicular from Wi to C and the line joining C to Mi and[Equation goes here - download the original to see it.] is the perpendicular distance. That is torque = force perpendicular distance results in [Equation goes here - download the original to see it.] But C is the centre of mass, so the sum of all the moments of the mass elements about this centre of mass is equal to 0. i.e. [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] We are given that the moment of inertia of p about z is Mk2. In terms of our mass elements Mi, distance ri from c this means [Equation goes here - download the original to see it.] Now consider the moment of inertia of p about another axis l' with perpendicular distance d from z. [Diagram goes here - download the original to see it.] We consider a mass element of Mi of p which has perpendicular distance ri from z and xi from z'. The moments of inertia of p about z' is approximately the sum of the moments of inertia of all the elements of p [Equation goes here - download the original to see it.] In the limit, as [Equation goes here - download the original to see it.], the volume of mi, tends to O, the approximation becomes exact [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] But [Equation goes here - download the original to see it.] Thus [Equation goes here - download the original to see it.]
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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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