Question

state theorem of parallel n perpendicular axes

Answer 1

This theorem is applicable only to the planar bodies. Bodies which are flat with very less or negligible thickness. This theorem states that the moment of inertia of a planar body about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with the perpendicular axis and lying in the plane of the body.



( Source: Toproadrunner5 )

In the above figure, we can see the perpendicular body. So Z axis is the axis which is perpendicular to the plane of the body and the other two axes lie in the plane of the body. So this theorem states that

IZ =  Ix  +  Iy

That means the moment of inertia about an axis which is perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes.

Let us see an example of this theorem:

Suppose we want to calculate the moment of inertia of a uniform ring about its diameter. Let its centre be MR²/2, where M is the mass and R is the radius. So, by the theorem of perpendicular axes, IZ =  Ix  +  Iy. Since the ring is uniform, all the diameters are equal.

∴  Ix  = Iy

∴ IZ  = 2 Ix

Iz = MR²/2

So finally the moment of inertia of a disc about any of its diameter is MR²/4

Learn more about Moment of Inertia in detail here

Parallel Axis Theorem

Parallel axis theorem is applicable to bodies of any shape. The theorem of parallel axis states that the moment of inertia of a body about an axis parallel to an axis passing through the 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 the distance between the two axes.

IZ’ = Iz + Mα²

where, α is the distance between two axes.

Solved Examples For You

Q1. The moment of inertia of a thin uniform rod of mass M and length L bout an axis perpendicular to the rod, through its centre is I. The moment of inertia of the rod about an axis perpendicular to the rod through its endpoint is:

I/4I/22I4I

Answer: D. Icentre =  ML²/12 and Iendpoint = ML²/3 = 4I

Answer 2

Theorem of Perpendicular Axis

This theorem states, “The sum of the moment of inertia of a laminar body about any two mutually perpendicular axes in the plane is equal to its moment of inertia about an axis perpendicular to its plane and passing through the point of intersection of the two axes.”

Let Ix, Iy and Iz be the moments of inertia of plane lamina three mutually perpendicular axes passing through the point O. OX and OY axes are in the plane and axis OZ is perpendicular to the plane, then

Iz=Ix+Iy

Proof

Let OX and OY be two mutually perpendicularly axes in the plane of the lamina and OZ be an axis passing through O and perpendicular to the plane of lamina. If a particle P of mass m is at a distance r from O, the moment of inertia of this particle about the axis OZ = mr2. The moment of inertia of the inertia of the entire body about the OZ-axis is given by

Iz=∑mr2

The moment of inertia of the body about OX-axis is

IX=∑my2and about OY-axis,Iy=∑mr2.then,Ix+IyIx+Iywhere r2=x2+y2therefore, from equation(i)and equation(ii),we haveIx+Iy=∑m(x2+y2)=∑mr2…(ii)=Iz…(iii)

Which proves the theorem of perpendicular axis for a laminar body.