Parallel Axis Theorem
Parallel Axis Theorem (PAT) in tensor notation is given by:
𝐼 = 𝐼0 + 𝑚[𝐸 3 ∗ 𝐷𝑜𝑡( 𝑅, 𝑅 ) − 𝑅 ⨂ 𝑅]
𝐼 is the final inertia tensor as transformed away from the origin; 𝐼0 is the initial inertia tensor; 𝑚 is the
mass of the object to be transformed; 𝐸3 is the identity matrix; 𝑅 is the vector (pointing towards or
away from the origin) to transform along; ⨂ represents the outer products of its left and right operands.
Hemisphere Inertia
The inertia of a hemisphere about the center of its base is given by, where 𝑚ℎ is half the mass of a
sphere:
𝐼ℎ = 𝐸 3 ∗ (2⁄5 𝑚ℎ ∗ 𝑟 2 )
This can be shifted towards the center of mass of the hemisphere by negating the PAT equation. 𝑅 is
0
3
given by { ⁄8 𝑟} :
0
𝐼ℎ = 𝐼ℎ0 − 𝑚ℎ [𝐸3 ∗ 𝑟 2 − 𝑅 ⨂ 𝑅]
Due to symmetry the 𝐼ℎ𝑥𝑥 element and 𝐼ℎ𝑧𝑧 elements are identical. In scalar form the 𝐼ℎ𝑥𝑥 element is
computed as:
𝐼ℎ𝑥𝑥 = 𝐼ℎ𝑧𝑧 = 𝐼ℎ − 𝑚ℎ ∗ (3⁄8 ∗ r)2
𝐼ℎ𝑦𝑦 is computed as:
2
2
𝐼ℎ𝑦𝑦 = 𝐼ℎ − 𝑚ℎ ∗ [(3⁄8 ∗ r) − (3⁄8 ∗ r) ]
Collection of terms simplifies to:
𝐼ℎ𝑦𝑦 = 𝐼ℎ
Capsule Inertia
𝑦
3⁄ 𝑟
8
𝑟
𝑥
𝑙
To calculate the inertia tensor of a hemisphere relative to the center of mass of a cylinder, the
hemisphere must be translated to its own center of mass along the 𝑦 axis, then along the 𝑦 axis by
1⁄ 𝑙 + 3⁄ 𝑟. The final inertia moments for rotating about the x and z axes of a capsule are given by
2
8
(where 𝐼𝑐 is the inertia of a cylinder):
2
2
𝐼𝑥𝑥 = 𝐼𝑧𝑧 = 𝐼𝑐𝑥𝑥 + 2𝐼ℎ𝑥𝑥 + 2𝑚ℎ [(3⁄8 𝑟 + 1⁄2 l) − (3⁄8 𝑟) ]
Which simplifies to:
(3𝑟 + 2𝑙)
𝐼𝑥𝑥 = 𝐼𝑧𝑧 = 𝐼𝑐 + 2𝐼ℎ + 2𝑚ℎ [
𝑙]
8