Question

write theorem of parallel axis of moment of inertia and prove it.​

Answer 1

Answer:

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Parallel Axis Theorem Derivation

Let Ic be the moment of inertia of an axis which is passing through the center of mass (AB from the figure) and I be the moment of inertia about the axis A’B’ at a distance of h.

Consider a particle of mass m at a distance r from the center of gravity of the body.

Then,

Distance from A’B’ = r + h

I = ∑m (r + h)2

I = ∑m (r2 + h2 + 2rh)

I = ∑mr2 + ∑mh2 + ∑2rh

I = Ic + h2∑m + 2h∑mr

I = Ic + Mh2 + 0

I = Ic + Mh2

Hence, the above is the formula of parallel axis theorem.

Parallel Axis Theorem of Rod

The parallel axis theorem of rod can be determined by finding the moment of inertia of rod.

Moment of inertia of rod is given as:

I = 13 ML2

The distance between the end of the rod and its center is given as:

h = L2.

Therefore, the parallel axis theorem of rod is:

Ic = 13ML2 – ML22

Ic = 13ML2 – 14ML2

Ic = 112 ML2.

Answer 2

Answer:

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.

Explanation:

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