wiz-icon
MyQuestionIcon
MyQuestionIcon
1
You visited us 1 times! Enjoying our articles? Unlock Full Access!
Question

Moment of inertia of a solid sphere of mass M and radius R is ______.


Open in App
Solution

Step 1. Given data:

Mass of solid sphere = M

Radius of solid sphere = R

Step 2. Calculations:

The moment of inertia of a solid sphere is defined as the quantity expressed by the body resisting angular acceleration which is equal to the sum of the product of the mass of every particle with its square of a distance from the axis of rotation.

Let us consider a sphere of radius R and mass M. A thin spherical shell of radius x, mass dm , and thickness dx is taken as a mass element. Volume density MV remains constant as the solid sphere is uniform throughout.

Volume V of solid sphere =43πR3

The volume of thin spherical shell element dv = 4πx2.dx

Now, the Volume density of the solid sphere = Volume density of the thin spherical shell element.

MV=dmdv

M43πR3 = dm4πx2dx

dm=M43×πR3×4πx2dx=3M/R3.x2.dx

Step 3: Calculate the moment of inertia

Now, we can calculate the moment of inertia by putting the value in the following equation:

I=di=(23)×dm.x2=233MR3.x4.dx=2MR3x4.dx

Limits: As x increases from 0 to R, the elemental shell covers all the spherical sphere.

I equals open parentheses bevelled fraction numerator 2 M over denominator R cubed end fraction close parentheses integral subscript 0 superscript R x to the power of 4. d x
space space equals open parentheses bevelled fraction numerator 2 M over denominator R cubed end fraction close parentheses. open parentheses R to the power of 5 over 5 close parentheses
space space space equals space fraction numerator 2 M R squared over denominator 5 end fraction

Therefore, the moment of inertia of a uniform solid sphere I=2MR25.


flag
Suggest Corrections
thumbs-up
6
Join BYJU'S Learning Program
similar_icon
Related Videos
thumbnail
lock
Physical Pendulum
PHYSICS
Watch in App
Join BYJU'S Learning Program
CrossIcon