Question

state and prove perpendicular axis and parallel axis theorem? ​

Answer 1

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.

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Answer 2

perpendicular axis theorem:

M.O.I of a 2-dimensional object about an axis passing perpendicularly from it is equal to the sum of the M.O.I of the object about 2 mutually perpendicular axes lying in the plane of the object.

According to the above definition of Perpendicular axis theorem can be written as,

Izz = Ixx + Iyy

PROOF:

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 + y2²

Put this value in the above equation

Izz = ∫ (x² + y²) . dA

Izz = ∫ x².dA + y².dA

Izz= Ixx + Iyy

Hence proved.  

Parallel axis theorem:

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

                                                 I = I₀ + Mh²

• I is the moment of inertia of the body

• I₀ is the moment of inertia about the center

• M is the mass of the body

• h2 is the square of the distance between the two axes

Parallel Axis Theorem Derivation

Let I₀ 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 = I₀ + h²∑m + 2h∑mr

I = I₀ + Mh² + 0

I = I₀ + Mh²

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

Hope this helps you !!