Moment of Inertia of a Sphere

on . Posted in Classical Mechanics

moment of inertia Sphere hollow 1moment of inertia Sphere solid 1This calculation is for the moment of inertia of a sphere.  There are three separate calculations:  a solid sphere, a hollow sphere and a hollow core sphere.  The difference between a hollow sphere and a hollow core sphere is a hollow sphere has a thin shell, or a thickness that is neglible.  Because a sphere is the same dimensions in every dimension, the moment of inertia is the same about every axis.

Moment of Inertia of a Sphere Index

 

Moment of Inertia of a Sphere Formula, Solid Sphere

\( I = \frac {2}{5} \; m \; r^2 \) 
Symbol English Metric
\(\large{ I }\) = moment of inertia \(lbm\;/\;ft^2-sec\) \(kg\;/\;m^2\)
\(\large{ m }\) = mass \( lbm \) \( kg \)
\(\large{ r }\) = radius \( in \) \( mm \)

 

Moment of Inertia of a Sphere Formula, Hollow Sphere

\( I = \frac {2}{3} \; m \; r^2 \) 
Symbol English Metric
\(\large{ I }\) = moment of inertia \(lbm\;/\;ft^2-sec\) \(kg\;/\;m^2\)
\(\large{ m }\) = mass \( lbm \) \( kg \)
\(\large{ r }\) = radius \( in \) \( mm \)

 

Moment of Inertia of a Sphere Formula, Hollow Core Sphere

\( I = \frac {2}{5} \; m \; \left( \; r_2^5 \;-\; r_1{^5} \;/\; r_2{^3} \;-\; r_1{^3}\;  \right)   \) 
Symbol English Metric
\(\large{ I }\) = moment of inertia \(lbm\;/\;ft^2-sec\) \(kg\;/\;m^2\)
\(\large{ m }\) = mass \( lbm \) \( kg \)
\(\large{ r }\) = radius \( in \) \( mm \)
\(\large{ r_1 }\) = radius \( in \) \( mm \)
\(\large{ r_2 }\) = radius \( in \) \( mm \)

 

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Tags: Moment of Inertia