Calculating the Moment of Inertia of a Capsule

A capsule can be decomposed into a cylinder and two hemispheres, where each hemisphere sits on a cylinder cap. The moment of inertia of a capsule can be calculated as a composition of the moment of inertia of a cylinder and the moment of inertia of a hemisphere.

Consider a capsule of radius $R$ and cylindrical length $L$ centered at the origin with its main axis (i.e. axis of rotational symmetry) aligned with the $x$ axis on a right-hand coordinate system where $y$ points up. The total length of the capsule is $L + 2R$ and its mass is $M$ .

The volume of the cylindrical portion is

$V_{cyl} = \pi R^2 L$

The volume of a hemisphere is

$V_{hemi} = \frac{2 \pi R^3}{3}$

The volume of the capsule is the sum

$V = V_{cyl} + 2 V_{hemi}$

Thus, the mass of the cylindrical portion is

$M_{cyl} = \frac{V_{cyl}}{V} M$

and the mass of a hemisphere is

$M_{hemi} = \frac{V_{hemi}}{V} M$

The moment of inertia of the cylindrical portion along the $x$ axis is:

$I_{cyl \, xx} = \frac {M_{cyl} R^2}{2}$

The moment of inertia of one of the hemispheres along the $x$ axis is:

$I_{hemi \, xx} = \frac{2}{5} M_{hemi} R^2$

Thus, the moment of inertia of the capsule along the $x$ axis is the sum:

$I_{xx} = I_{cyl \, xx} + 2 I_{hemi \, xx} = \frac {M_{cyl} R^2}{2} + \frac{4}{5} M_{hemi} R^2 = \frac{\left( 5 M_{cyl} + 8 M_{hemi} \right) R^2}{10}$

The moment of inertia of a cylinder along the $z$ or $y$ axis is

$I_{cyl \, yy} = I_{cyl \, zz} = \frac{M_{cyl}}{12} \left( L^2 + 3 R^2 \right)$

The moment of inertia of a hemisphere along the $z$ or $y$ axis is

$I_{hemi \, yy} = I_{hemi \, zz} = \frac{2}{5} M_{hemi} R^2$

The moment of inertia of the capsule along the $y$ and $z$ axes are the same due to symmetry. Let’s calculate $I_{zz}$ using the parallel axis theorem to add the moment of inertia of the hemispheres whose center of mass is at a distance of $L/2 + 3R/8$ from the origin.

$I_{zz} = I_{cyl \, zz} + 2 \left( I_{hemi \, zz} + M_{hemi} \left( \frac{L}{2} + \frac{3}{8} R \right)^2 \right)$

$= I_{cyl \, zz} + 2 I_{hemi \, zz} + M_{hemi} \frac{(4L + 3R)^2}{32}$

Likewise

$I_{yy} = I_{cyl \, yy} + 2 I_{hemi \, yy} + M_{hemi} \frac{(4L + 3R)^2}{32}$