Counterintuitive Result Regarding COM Of A Hemispherical Shell

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Discussion Overview

The discussion revolves around the center of mass (COM) of a hemispherical shell, exploring the counterintuitive results when comparing it to the COM of a semicircular wire. Participants examine the implications of different shapes and densities on the location of the COM.

Discussion Character

  • Exploratory, Debate/contested, Conceptual clarification

Main Points Raised

  • One participant describes a process involving a semicircular wire and its COM, suggesting that the COM remains unchanged through various rotations.
  • Another participant notes that the wire's shape differs from that of a wedge, implying that this affects the COM calculation.
  • A different perspective is offered regarding the finite thickness of wires, suggesting that mass accumulation at the center of the hemisphere leads to a higher density there compared to the edges.
  • One participant expresses a desire for a mathematical approach to calculating the COM of a hemisphere using integrals.
  • Another participant reflects on the cumulative nature of the semicircular wire elements and their respective COMs, indicating a puzzling discrepancy between individual and cumulative COM locations.

Areas of Agreement / Disagreement

Participants express differing views on the factors affecting the COM of the hemispherical shell, indicating that multiple competing perspectives remain without a clear consensus.

Contextual Notes

Participants have not resolved the implications of wire thickness and shape on the COM, nor have they clarified the mathematical steps necessary for calculating the COM of a hemisphere.

Who May Find This Useful

This discussion may be of interest to those studying physics, particularly in areas related to mechanics and center of mass calculations.

GPhab
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Follow the below four steps to amuse yourself

1) Take a semicircular wire. You know that its COM is at (0,2R/\pi). Now pass an axis through its COM and perpendicular to the line joining its ends.

2)Rotate this "half-lollipop" about the axis fixed till it comes to another position. The COM obviously didn't undergo any displacement.

3)Do this is an umpteen number of times and imagine as if a new wire is created for each position. The COMs of all of these wires coincide and should be at the coordinate mentioned in step 1 (Including a Z-co-ordinate, which can be taken as zero)

4)But the COM of a hemispherical shell is R/2 above the centre!

COM-centre of mass
 
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because the wire isn't shaped like a wedge.
 
If the wires have finite thickness, then there's an accumulation of mass at the center of the hemisphere where multiple wires would cross. The density at the center (top) of the hemisphere would be higher than at the edges, which differs from a shell of finite thickness.
 
Jeff got it right. To what extent did it amuse you?
 
Speaking of which, can anyone show me how to calculate the COM of a hemisphere (integral way)?
 
GPhab said:
Jeff got it right. To what extent did it amuse you?

We assume that the hemispherical shell is being built up from the sum of semicircular wire. Each element of the semicircular wire has COM at the point A. But the cummulative sum of the elements has COM at B. That puzzling :confused: