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Rotation & Angular Dynamics · Physics

Shift axes without redoing the whole integral.

Axis theorems let you compute moments of inertia about new axes quickly. Learn when each theorem applies and how to avoid common “wrong axis” mistakes.

This topic

Parallel and Perpendicular Axis Theorems

Use the parallel-axis theorem to shift axes, and the perpendicular-axis theorem to relate planar inertias.

Shift
Parallel-axis theorem
The parallel-axis theorem relates the moment of inertia about any axis parallel to one through the center of mass.
  • Statement: I = ICM + Md²
  • d is the distance between the parallel axes
  • Why “through CM” matters
  • Typical use: rod about end, disk about tangent axis
Start nano-lesson Make sure your axis is actually parallel.
Planar
Perpendicular-axis theorem
For a flat (planar) body, the moment of inertia about an axis perpendicular to the plane is the sum of the moments about two perpendicular axes in the plane.
  • Statement: Iz = Ix + Iy (planar lamina)
  • Axes must intersect at the same point
  • Works only when mass is in a plane
  • Useful for plates and thin disks
Start nano-lesson Check the “thin, planar” requirement.
How to use
Choosing the right theorem
Many inertia mistakes come from using a theorem outside its conditions. Learn a quick checklist to decide which tool applies.
  • Parallel shift? Use parallel-axis theorem
  • Need I about z for a flat lamina? Use perpendicular-axis theorem
  • Axes must be clearly defined in words and diagrams
  • Know what is given: ICM vs I about some other axis
Start nano-lesson Write the axis names before any math.
Worked habits
Common applications and pitfalls
These theorems are fast when you set up the geometry cleanly. Most errors come from mixing up the distance d, the intersection point, or the planar requirement.
  • Rod: center axis → end axis via d = L/2
  • Disk: center axis → tangent axis via d = R
  • Perpendicular-axis: x and y must intersect at the same point
  • Do not use perpendicular-axis for 3D solids
Start nano-lesson Geometry first, theorem second.
Practice
Practice & Exercises
Practice shifting axes with the parallel-axis theorem and using the perpendicular-axis theorem for planar shapes.
  • Compute I about a new parallel axis using ICM + Md²
  • Apply perpendicular-axis theorem to thin plates
  • Identify when a theorem is not allowed
  • Multi-step “find I then use τ = Iα” prompts
  • Exam-style axis-theorem sets
Open practice sets Label axes clearly in every solution.