Inertia Formula for Hollow Cylinder

Aug 2, 2024

Deriving the Formula for the Inertia of a Hollow Cylinder

Key Definitions and Setup

  • Inner Radius (R1): Radius of the inside part of the hollow cylinder.
  • Outer Radius (R2): Radius of the outer part of the hollow cylinder.
  • Axis of Rotation: Axis about which the cylinder rotates.
  • Generic Point (r): Distance between the axis of rotation and a point within the cylinder.
  • Length (L): Length of the cylinder.

Volume and Density

  • Volume of a cylinder: (V = \pi r^2 L)
  • Density (\rho): Mass over volume (\rho = \frac{M}{V})
  • Differential Mass (dM): (dM = \rho dV)
  • Differential Volume (dV): (dV = 2\pi r L dr)

Inertia Calculation

  1. Substituting (dV) into (dM): [ dM = \rho \cdot 2\pi r L dr ]

  2. Inertia formula: [ I = \int_{R1}^{R2} r^2 dM ]

  3. Substitute (dM) into the inertia formula: [ I = \int_{R1}^{R2} r^2 \cdot \rho \cdot 2\pi r L dr ]

  4. Moving constants out of the integral: [ I = 2\pi \rho L \int_{R1}^{R2} r^3 dr ]

  5. Evaluating the integral: [ \int_{R1}^{R2} r^3 dr = \left. \frac{r^4}{4} \right|_{R1}^{R2} = \frac{R2^4}{4} - \frac{R1^4}{4} ]

  6. Substituting back into the inertia equation: [ I = \frac{2\pi \rho L}{4} (R2^4 - R1^4) ]

  7. Simplifying: [ I = \frac{\pi \rho L}{2} (R2^4 - R1^4) ]

Volume of a Hollow Cylinder

  • Volume of large cylinder (V2): (\pi R2^2 L)
  • Volume of small cylinder (V1): (\pi R1^2 L)
  • Volume of hollow cylinder: (V = V2 - V1 = \pi (R2^2 - R1^2) L)

Mass of the Hollow Cylinder

  • Mass (M): (M = \rho V)
  • Substitute volume: [ M = \rho \pi (R2^2 - R1^2) L ]

Final Inertia Formula

  1. Substituting the mass into the inertia equation: [ I = \frac{\pi \rho L}{2} (R2^4 - R1^4) = \frac{M}{\pi \rho (R2^2 - R1^2) L} \cdot \frac{\pi \rho L}{2} (R2^4 - R1^4) ]

  2. Simplifying: [ I = \frac{M}{2} (R2^2 + R1^2) ]

Inertia for a Solid Cylinder

  • For a solid cylinder, inner radius (R1) is 0: [ I = \frac{M}{2} R2^2 ]
  1. If there is only one radius, we drop the subscript 2: [ I = \frac{M}{2} R^2 ]

Summary

  • Inertia of a hollow cylinder: (I = \frac{M}{2} (R2^2 + R1^2))
  • Inertia of a solid cylinder: (I = \frac{M}{2} R^2)