Subject
Level
College
Year
1
Lesson Steps:

Note: I'm using Iᶻ, Iˣ, Iʸ because subscripts of y and z are not available to type here. use Iₓ, Iy and Iz

Statement:
The moment of inertia of a plane lamina about an axis perpendicular to its plane is equal to the sum of the moments of inertia of the lamina about two axes perpendicular to each other in its plane intersecting each other at the point through where the perpendicular axis passes.
Iᶻ = Iˣ + Iʸ

Proof:
Consider a particle of mass 'm' at 'p' in the plane lamina. Let X and Y-axis lie in the plane of the lamina and Z-axis be perpendicular to the lamina. Let the particle be at a distance 'r' from the Z-axis. Let the moments of inertia of the plane lamina about X, Y, Z axes be Iˣ, Iʸ and Iᶻ respectively.

So, Iˣ = Σmy² and Iʸ = Σmx²

Iᶻ = Σmr² = Σm(x²+y²) [because r²=x²+y²]

So,

Iᶻ = Iˣ + Iʸ

which is the theorem.

2 (b) Refer the attached image.

2 (a) Refer to the steps
(b)1/√2

Level: College
Subject: Chemistry
Topic: Chemical bonding