Rankwise

Moment of inertia of a thin hollow cylinder about its own axis

I = MR², radius of gyration k = R

Axis: about its own axis.

kg
m

Derivation

  1. All the mass sits at distance R from the axis, like a stack of rings.
  2. I = MR².

Moment of inertia of standard bodies

Body and axisI
Thin ring, through the centre, perpendicular to its planeMR²
Thin ring, about a diameterMR²/2
Thin ring, about a tangent in its plane3MR²/2
Thin ring, about a tangent perpendicular to its plane2MR²
Uniform disc, through the centre, perpendicular to its planeMR²/2
Uniform disc, about a diameterMR²/4
Uniform disc, about a tangent in its plane5MR²/4
Uniform disc, about a tangent perpendicular to its plane3MR²/2
Thin rod, through the centre, perpendicular to its lengthML²/12
Thin rod, through one end, perpendicular to its lengthML²/3
Solid sphere, about a diameter2MR²/5
Thin hollow sphere (spherical shell), about a diameter2MR²/3
Solid sphere, about a tangent7MR²/5
Thin hollow sphere, about a tangent5MR²/3
Solid cylinder, about its own axisMR²/2
Thin hollow cylinder, about its own axisMR²
Solid cylinder, through the centre, perpendicular to its axisM(R²/4 + L²/12)
Rectangular plate (sides a and b), through the centre, perpendicular to the plateM(a² + b²)/12
Annular disc (radii R₁ and R₂), through the centre, perpendicular to its planeM(R₁² + R₂²)/2
Solid cone, about its own axis3MR²/10

Questions students ask

What is the moment of inertia of a thin hollow cylinder about its own axis?

I = MR², so the radius of gyration is k = R.

How is the moment of inertia of a thin hollow cylinder about its own axis derived?

All the mass sits at distance R from the axis, like a stack of rings. I = MR².

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