Question

explain perpendicular axis therom and parallel axis therom
$explain perpendicular axis therom and parallel axis therom$
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Answer 1

Answer:

●The parallel axis theorem states that

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

Parallel Axis Theorem Derivation

Let Ic be the moment of inertia of an axis that is passing through the centre of mass (AB from the figure) and I will 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 centre 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

●Perpendicular axis theorem states that

For any plane body the moment of inertia about any of its axes which are perpendicular to the plane is equal to the sum of the moment of inertia about any two perpendicular axes in the plane of the body which intersect the first axis in the plane.

Let us assume there are three mutually perpendicular axes named as X, Y and Z. They are meeting at origin O.

Now consider the object lies in the XY plane having a small area dA. It is having y distance from X-axis and x distance from Y-axis. Its distance from the origin is r.

 

Let IZ, IX and IY be moments of Inertia about the X, Y and Z axis respectively.

Moment of Inertia about Z-axis i.e.

IZ = ∫ r².dA …………. (i)

Here, r²= x² + y2

Put this value in the above equation

IZZ = ∫ (x²+ y²) . dA

IZZ = ∫ x².dA + y².dA

IZZ = IXX + IYY

Hence proved.

☆ I HOPE ITS HELP YOU ☆

Answer 2

Answer:

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