## (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.

Mechanics with vectors, derivatives and integrals.

Level 3 requires the basics of vector calculus, differential and integral calculus. Suitable for undergraduates and high school students.

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Derivation ## (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

## Related formulas

Formula ## Thin circular ring (moment of inertia)

`$$ \class{brown}{I} ~=~ \class{brown}{m} \, r^2 $$`Formula ## Moment of Inertia - Sphere (Axis of Rotation Through the Center)

`$$ \class{brown}{I} ~=~ \frac{2}{5} \, \class{brown}{m} \, r^2 $$`Formula ## Moment of Inertia - Cuboid with Axis of Rotation Through the Center Point

`$$ \class{brown}{I} ~=~ \frac{\class{brown}{m}}{12} \, \left( l^2 + w^2 \right) $$`Formula ## Moment of Inertia - Hollow Cylinder (Axis of Rotation Parallel to Radius)

`$$ \class{brown}{I} ~=~ \frac{1}{2} \, \class{brown}{m} \, \left( r^2 ~+~ \frac{w^2}{6} \right) $$`Formula ## Solid cylinder - axis of symmetry (moment of inertia)

`$$ \class{brown}{I} ~=~ \frac{1}{2} \, m \, {\class{purple}{r}}^2 $$`Formula ## Moment of Inertia - Solid Cylinder (Rotation Perpendicular to Symmetry Axis)

`$$ \class{brown}{I} ~=~ \frac{1}{12} \, \class{brown}{m} \, l^2 $$`Formula ## Steiner's Theorem (Moment of Inertia)

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

Derivation ## 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.

## Related Illustrations