The x,y,z coordinates of CM of a solid cylinder

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Discussion Overview

The discussion centers around determining the x, y, and z coordinates of the center of mass (CM) of a solid cylinder, with a focus on integration methods and geometrical considerations. The scope includes theoretical reasoning and mathematical calculations related to the centroid of cylindrical shapes.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents calculated coordinates for the CM of a solid cylinder, stating specific values for x, y, and z based on integration.
  • Another participant suggests that for a complete cylinder, the CM is at the midpoint, while for a quadrant, the y coordinate is confirmed as h/2.
  • A later reply proposes that the CM should lie on the plane of symmetry at 45 degrees, implying that the x and y offsets should be equal.
  • One participant mentions the availability of centroid values for cylindrical segments on Wolfram Alpha and references the complexity of the quarter tank problem.
  • Another participant agrees with the initial calculations and suggests a simpler method for finding the centroid of a semicircle.

Areas of Agreement / Disagreement

Participants express varying degrees of agreement regarding the correctness of the calculated coordinates, with some affirming the values presented while others suggest alternative methods or considerations. The discussion does not reach a consensus on the final correctness of the calculations.

Contextual Notes

There are references to specific geometrical configurations (complete cylinder vs. quadrant) and assumptions about the placement of the origin and axes, which may affect the calculations. The discussion also highlights the need for additional planes in certain problems, indicating potential limitations in the presented solutions.

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I have found via integration that the y coordinate is $$y =h/2 = 120 mm$$. The x coordinate is $$x = \frac{-4r}{3\pi} = -51.9mm$$ and the z coordinate is $$z = r - \frac{4r}{3\pi} = 69.1 mm$$. I have no answers in my textbook so can't confirm whether i am correct or not.
 
Last edited:
Engineering news on Phys.org
Google is you friend here.
 
If the cylinder is complete then the centre of mass is the midpoint.

If it is a quadrant of a solid cylinder as shown in your picture then it will be at C.y = h/2 as you show. The CM will lie on the plane of symmetry at 45 degrees, so the C.x and C.y offsets from the cylinder axis should have identical values. I suspect your values are correct.
Would it not be easier if you placed your origin and z axis on the cylinder axis ?

The centroid of a cylindrical segment, hoof or wedge is available on Wolfram Alpha. http://mathworld.wolfram.com/CylindricalSegment.html
Unfortunately the quarter tank problem has only one plane cut. You need two.
But one of the values will be the same as for a half cylinder which has a single plane cut.
 
Your values appear to be correct. Attached is a one page extract from an 8MByte .pdf

There is an even easier solution.
The centroid of a semicircle is R * 4 / 3π
 

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