Moment of Inertia of a Solid Cylinder With a Wedge Removed

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SUMMARY

The discussion focuses on calculating the moment of inertia of a solid cylinder with a wedge removed. Participants clarify the importance of specifying the axis of rotation, particularly whether it is the z-axis through the origin or through the mass center. The formula presented, r² = ρ² + z², is identified as incorrect for the context of this calculation. Accurate definitions and formulas are essential for determining the moment of inertia in this scenario.

PREREQUISITES
  • Understanding of moment of inertia concepts
  • Familiarity with solid geometry, specifically cylinders
  • Knowledge of rotational axes in physics
  • Basic proficiency in mathematical notation and calculations
NEXT STEPS
  • Study the derivation of the moment of inertia for composite shapes
  • Learn about the parallel axis theorem in physics
  • Explore the application of integral calculus in calculating moments of inertia
  • Review solid mechanics principles related to rotational dynamics
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Physics students, mechanical engineers, and anyone involved in the study of rotational motion and dynamics will benefit from this discussion.

cedoty1989
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Homework Statement
Imagine a solid cylinder height h able to rotate vertically around z-axis (centered at x=0 y=0). There is a wedge cut out so that when looking down with z hat pointing out of the page there is an angle 2a formed such that their is symmetry with respect to reflection over the x axis (See picture). The question is to calculate the moment of inertia.
Relevant Equations
I=∫∫∫ dm r^2 -> Cylindrical Coords: (r from 0 to R) (z from zero to h) (theta from -Pi + a to Pi-a)

Uniform density -> dm=(M/V)dV

dV = (dr)r(dtheta)(dz)

The rest of the equations are in the picture...
CamScanner 11-23-2020 21.50-1.jpg
 

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I know there should be an alpha in there somewhere but I cannot see where I'm going wrong. Thank you in advance.
 
cedoty1989 said:
The question is to calculate the moment of inertia.
About what axis? z axis through origin or through mass centre?
Your r2=ρ2+z2 doesn't make sense for either.
 

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