Moment of Inertia Sphere

The moment of inertia of a point mass about a known axis is defined by I = mr²  where m is its mass and r is its perpendicular distance from the axis of rotation.

Introduction :

Definition :  The moment of inertia of a rigid body about an axis is defined as the sum of the products of the masses of different particles, supposed to be constituting the body, and the square of their respective perpendicular distances from the axis of rotation.
                     Moment of inertia of a sphere can be explained in two parts (1) Solid Sphere (2)Hollow Sphere.
(1) Moment of inertia of a Solid Sphere  :  
    (a) About an axis passing through its diameter :   Consider a solid sphere of mass M and radius R. Its moment of inertia about an axis of rotation passing through its diameter is
                                                                            I  =  MR²
   (b) About an axis passing through its tangent  :   Let A'B' the tangent to the solid sphere. A parallel axis through its centre of mass is AB

By parallel axes theorem,
 Moment of inertia about the tangent = Moment of inertia about a diameter + Mr² .
                                                                  =`(2)/(5)` MR² + MR²
                                                         I      =  `(7)/(5)` MR² .

Moment of Inertia Sphere : Hollow Sphere

(2) Moment of Inertia of a Hollow Sphere
(a) Moment of inertia about an axis passing through the diameter of a hollow sphere of mass M and radius R is
                                                I  =`(2)/(3)`   MR² . 
(b)  Moment of inertia about an axis passing through its tangent can be obtained by applying parallel axes theorem. It is given by
                                                  I  =  `(5)/(3)` MR² .

Moment of Inertia Sphere : Example Problem

Problem : If the radius of the earth is suddenly halved keeping its mass constant, find its time period of rotation around its own axis.
Solution :  When the radius of the earth gets reduced suddenly keeping its mass constant, the angular momentum of the earth remains constant.      
                                                I   = constant
     If I changes from I₁ to I₂ ,   changes from`omega` ₁  to`omega`  ₂ so that 
                                          I₁ `omega`₁  =  I₂`omega`₂ .
     Assuming the earth to be a solid sphere, its moment of inertia about its diameter,
                      I  =  MR²
   If the radius changes form R to R/n
                                           `(2)/(5)` MR₁² `omega`₁  =  `(2)/(5)` MR₂² `omega`₂ 
                                                           R²  =  `(2pi)/(24 hours)`  = [R / n]² `(2pi)/(T)`  
          The time period of rotation , T = 24 hours / n² .
      In this problem, the radius changes from  R  to R / 2 .
                         `:.`   `(2)/(5)` MR²  `(2pi)/(24 hours)`   =  `(2)/(5)` M [R/2]² `(2pi)/(T)`
                               T  =  24 / 2²   =  24 / 4  =  6 hours.