Uniform Thin Hoop Rotational Inertia Derivation

I = Σ m_i r_i

I = Σ m_i (r_i)^2

I = m r^2

I = m v^2

They are opposite concepts.

They describe different types of motion.

They are two different names for the same physical quantity.

Moment of Inertia is for linear motion, Rotational Inertia is for angular motion.

An object that can easily change its shape.

A point particle with negligible mass.

An object that does not change shape easily and is larger than a point particle.

Any object that is in constant motion.

"Uniform" means it has a constant radius, and "thin" means it has a small mass.

"Uniform" means it has constant density, and "thin" means its thickness is negligible compared to its radius.

"Uniform" means it is perfectly circular, and "thin" means it is made of a lightweight material.

"Uniform" means it has a smooth surface, and "thin" means it has a small circumference.

Its mass

Its density

Its thickness

Its radius

MR²

1/2 MR²

2/5 MR²

1/12 ML²

A solid cylinder

A solid sphere

A single particle with total mass M at radius R

A thin rod rotating about its center

Constants

Variables

The total radius and total mass

The axis of rotation and the density