Mechanics
Parallel Axis Theorem
Relates moment of inertia about any axis to the moment of inertia about a parallel axis through the center of mass.
Variables
Ikg·m²
Moment of inertia
About parallel axis
I_cmkg·m²
Moment of inertia (CM)
About center of mass
Mkg
Mass
Total mass
dm
Distance
Distance from CM to new axis
Derivation
Expanding I = ∫(r + d)²dm and using the definition of center of mass gives I = I_cm + Md².
Your notes
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Common questions
What is the Parallel Axis Theorem formula?
Parallel Axis Theorem is written I = Icₘ + Md². Relates moment of inertia about any axis to the moment of inertia about a parallel axis through the center of mass. It belongs to Mechanics, under Rotational Motion.
What are the units in I = Icₘ + Md²?
In SI units, I is moment of inertia, measured in kg·m²; I_cm is moment of inertia (cm), measured in kg·m²; M is mass, measured in kg; d is distance, measured in m.
How is the Parallel Axis Theorem formula derived?
Expanding I = ∫(r + d)²dm and using the definition of center of mass gives I = I_cm + Md². The result is I = Icₘ + Md².