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

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

 

Moment of Inertia of a Sphere Formula, Hollow Sphere

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

 

Moment of Inertia of a Sphere Formula, Hollow Core Sphere

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

 

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Tags: Moment of Inertia