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Prove the theorem of parallel axes....

Prove the theorem of parallel axes.

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Statement: "The moment of inertia of rigid body about any axis is equal to its moment of inertia about a parallel axes passing through its centre of mass plus product of mass of the body and the square of the perpendicular distance between the axes.

`I_(AB) = I_C +Mh^2`
where `I_(AB)` = Moment of inertia about AB axes.
`I_C` = Moment of inertia about the axes passing through the centre.
M = Total mass of the body.
h = Perpendicular distance between the two axes.
Proof: Let AB be axis about which moment of inertia is to be calculated. DЕ be the axis through centre of mass. Now consider a particle of mass m at F at distance x from C. Then `I_C` = moment of inertia of m about DE axis = `mx^2` .
`I_C` = Total moment of inertia of body = `Emx^2`
Moment of inertia of body about AB axis is
`I_(AB) = summ(x+h)^2`
`= summ(x^2+h^2+2hx)`
`I_(AB) = sumx^2+summh^2`
Here 2 hx term is dropped, because about axis mass has zero moment.
Also `summ` = M = Total mass of body and `I_C = summx^2`
Therefore, `I_(AB) = I_C+ Mh^2` is the required proof for parallel axis theorem.
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