Advanced Classical Mechanics

1

Newton's Laws of Motion

Newton's laws of motion describe the fundamental principles of how objects behave in response to forces. These laws explain the relationship between forces, mass, and acceleration.

2

(Hollow) Cylinder - Moment of Inertia

Derivation of the moment of inertia of a homogeneous hollow cylinder and a solid cylinder rotating around its axis of symmetry.

Questions & Answers

How to Calculate the Moment of Inertia of a Rigid Body? How to Calculate Moment of Inertia when the Axis of Rotation does not Pass Through the Center of Mass?

Related formulas

Thin circular ring (moment of inertia)

$$ \class{brown}{I} ~=~ \class{brown}{m} \, r^2 $$

Moment of Inertia - Sphere (Axis of Rotation Through the Center)

$$ \class{brown}{I} ~=~ \frac{2}{5} \, \class{brown}{m} \, r^2 $$

Moment of Inertia - Cuboid with Axis of Rotation Through the Center Point

$$ \class{brown}{I} ~=~ \frac{\class{brown}{m}}{12} \, \left( l^2 + w^2 \right) $$

Moment of Inertia - Hollow Cylinder (Axis of Rotation Parallel to Radius)

$$ \class{brown}{I} ~=~ \frac{1}{2} \, \class{brown}{m} \, \left( r^2 ~+~ \frac{w^2}{6} \right) $$

Solid cylinder - axis of symmetry (moment of inertia)

$$ \class{brown}{I} ~=~ \frac{1}{2} \, m \, {\class{purple}{r}}^2 $$

Moment of Inertia - Thin Rod (Rotation Perpendicular to Symmetry Axis)

$$ \class{brown}{I} ~=~ \frac{1}{12} \, \class{brown}{m} \, l^2 $$

Steiner's Theorem (Moment of Inertia)

$$ I ~=~ I_{\text{CM}} ~+~ \class{brown}{m} \, h^2 $$

3

Euler-Lagrange Equation and How to Use it in 5 Easy Steps

Here you will learn what the Euler-Lagrange equation is good for and how to apply it to concrete problems (for example, a thrown particle).

Derivations & Experiments

Euler-Lagrange Equation in 13 Steps

Here you learn the derivation of the Euler-Lagrange equation, which a function q(t) must satisfy for a functional S[q] to become stationary.

Course: Advanced Classical Mechanics