Moment of inertia of a cylinder?

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Homework Help Overview

The discussion revolves around calculating the moment of inertia of a cylinder with specific dimensions and mass density about a non-standard axis using multiple integration techniques.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to set up the integral using cylindrical coordinates but encounters difficulties with the assumptions regarding the distance from the axis of rotation. Some participants question the correctness of these assumptions and suggest clarifying the orientation of the cylinder.

Discussion Status

Participants are actively engaging in clarifying the setup of the problem, particularly the axis of rotation and the implications for the distance calculations. There is a focus on ensuring that the assumptions made about the geometry are accurate, with suggestions for providing additional details or sketches to aid understanding.

Contextual Notes

There is an indication that the axis of rotation is not aligned with the z-axis, which complicates the calculation of the moment of inertia. The need for clarity on the cylinder's orientation and the axis of symmetry is emphasized.

MoAli
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Homework Statement


the moment of inertia of A cylinder of height 2h radius (a) and uniform mass density ρ about a line x=y=z using multiple integration.

Homework Equations


I=ρ∫s^2*dV where the integral is over the volume V of cylinder and s is the perpendicular distance to the axis of rotation.

The Attempt at a Solution


I tried setting s^2=r^2+z^2 and integrate using cylindrical polars with elemental volume dV=rdrdθdz where r from 0 to a, z from -h to h, θ from 0 to 2π But I got the wrong answer.
 
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Check your assumptions about the distance of the mass element from the axis of rotation. If you're using cylindrical polar coordinates it should simply be the radius coordinate, no Pythagoras involved.
 
gneill said:
Check your assumptions about the distance of the mass element from the axis of rotation. If you're using cylindrical polar coordinates it should simply be the radius coordinate, no Pythagoras involved.
well, that still is a wrong answer, the thing is the line about which the rotation occurs is not z axis, if it was then yes perpendicular distance is the radius.
 
MoAli said:
well, that still is a wrong answer, the thing is the line about which the rotation occurs is not z axis, if it was then yes perpendicular distance is the radius.
The you'll have to be specific about the orientation of the cylinder and which axis forms the axis of symmetry. Post a sketch if you can, and show us the details of your attempt.
 

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