Question

derive an expression of parallel aces theorem
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Answer 1

The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to axis passing through centre of mass is equal to the sum of the moment of inertia of body about an axis passing through centre of mass and product of mass and square of distance between the two axes.

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Answer 2

The formula of parallel axis theorem is I = Ic  + $Mh^{2}$ .

Parallel axis theorem :

The moment of inertia of a body about an axis which is parallel to the body passing through its centre and it is equal to the sum of the moment of inertia of the body about the axis which is passing through the centre and the product of the mass of the body and square of the distance of between two axes.

It can be expressed as:

I = Ic + M$h^{2}$

Where,

The moment of inertia about the center= Ic

The mass of the body= M

The square of the distance = $h^{2}$

Derivation:

Let moment of inertia of an axis be Ic which is passed through the centre of mass and m be the mass a particle ,r is distance.

Then,

Distance= r + h

I = ∑m $(r + h)^{2}$

I = ∑m ($r^{2}$ + $h^{2}$  + 2rh)

I = ∑$mr^{2}$ + ∑$mh^{2}$ + ∑2rh

I = Ic +∑$h^{2}$m + 2h∑mr

I = Ic + $Mh^{2}$ + 0

I = Ic  + $Mh^{2}$

Hence, the formula of parallel axis theorem is I = Ic  + $Mh^{2}$ .

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