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
Vcyl=πR2LThe volume of a hemisphere is
Vhemi=2πR33The volume of the capsule is the sum
V=Vcyl+2VhemiThus, the mass of the cylindrical portion is
Mcyl=VcylVMand the mass of a hemisphere is
Mhemi=VhemiVMThe moment of inertia of the cylindrical portion along the x axis is:
Icylxx=McylR22The moment of inertia of one of the hemispheres along the x axis is:
Ihemixx=25MhemiR2Thus, the moment of inertia of the capsule along the x axis is the sum:
Ixx=Icylxx+2Ihemixx=McylR22+45MhemiR2=(5Mcyl+8Mhemi)R210The moment of inertia of a cylinder along the z or y axis is
Icylyy=Icylzz=Mcyl12(L2+3R2)The moment of inertia of a hemisphere along the z or y axis is
Ihemiyy=Ihemizz=25MhemiR2The moment of inertia of the capsule along the y and z axes are the same due to symmetry. Let’s calculate Izz 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.
Izz=Icylzz+2(Ihemizz+Mhemi(L2+38R)2) =Icylzz+2Ihemizz+Mhemi(4L+3R)232Likewise
Iyy=Icylyy+2Ihemiyy+Mhemi(4L+3R)232