Perpendicular-Axis Theorem
For a thin lamina lying in a plane, the moment of inertia about an axis perpendicular to the plane through a point equals the sum about any two mutually perpendicular in-plane axes through that point.
Why this shows up in the exam
Relating disc diameter and central-axis inertia · Finding lamina inertia about diagonals · Combining rectangular-sheet axes
Learn the idea
For a planar lamina, inertia about the perpendicular axis equals the sum about two perpendicular in-plane axes. For a planar lamina, inertia about the perpendicular axis equals the sum about two perpendicular in-plane axes. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: For a flat body, out-of-plane equals the two in-plane inertias.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- I_z = I_x + I_y — Perpendicular-axis theorem for a planar lamina with concurrent axes.
How to approach it
- 1Confirm the body is planar
- 2Choose concurrent perpendicular x and y axes
- 3Add their inertias for the perpendicular z axis
Common slip-ups that cost marks
- •Applying the theorem to a three-dimensional body
- •Using axes through different points
- •Assuming I_x equals I_y without symmetry
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
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