description
 Statement of Parallel and Perpendicular Axes Theorems and Their Applications
 Theorem of perpendicular axes
 Theorem of parallel axes
notes
Theorem of perpendicular axis:
Perpendicular Axis Theorem: The moment of inertia of a planar body (lamina) about an axis perpendicular to its plane is equal to the sum of its moments of inertia about two perpendicular axes concurrent with perpendicular axis and lying in the plane of the body.
Moment of inertia about xaxis, I_{x }= mx^{2}
Moment of inertia about yaxis, I_{y }= my^{2}
Moment of inertia about zaxis, `"I"_z= m(sqrt(x2+y2))^2`
`I_x + I_y = mx^2 + my^2`
= `m(x^2 + y^2)`
=`m(sqrt(x2+y2))`
Ix+Iy=Iz
Hence the theorem is proved
Applicable only to planar bodies.
Theorem of parallel axis:
Parallel Axis Theorem: The moment of inertia of a body about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing through its centre of mass and the product of its mass and the square of the distance between the two parallel axes.
This theorem is applicable to a body of any shape.
Example: Given the moment of inertia of a disc of mass M and radius R about any of its diameters to be `"MR"^2/4`, find its moment of inertia about an axis normal to the disc and passing through a point on its edge
Solution:
We can apply Perpendicular axis theorem here on x axis & y axis and get I’, moment of inertia in z axis.
`"l"_"z" = "l"_"x" + "l"_"y"`, now as because of symmetry I_{x} & I_{y} are same so I_{z} = I’ = 2I = `("Mr"^2)/2`
Now we can apply parallel axis theorem to find I’’.
`I'' = I' + "MR"^2 = 3/2("MR"^2)`
Moment of Inertia:
Z 
Body 
Axis 
Figure 
I 
1 
Thin circular ring, radius R 
Perpendicular to plane, at centre 
MR^{2} 

2 
Thin circular ring, radius R 
Diameter 
`"MR"^2/2` 

3 
Thin rod, length L 
Perpendicular to rod, at mid point 
`"ML"^2/12` 

4 
Circular disc, radius R 
Perpendicular to disc at centre 
`"MR"^2/2` 

5 
Circular disc, radius R 
Diameter 
`"MR"^2/4` 

6 
Hollow cylinder, radius R 
Axis of cylinder 
MR^{2} 

7 
Solid cylinder, radius R 
Axis of cylinder 
`"MR"^2/2` 

8 
Solid sphere, radius R 
Diameter 
2 `"MR"^2/5` 