Finding the Centre of Mass and Toppling Point of a Spinning Top

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Discussion Overview

The discussion revolves around finding the center of mass of a uniform solid spinning top shaped as an inverted right circular cone with a cylindrical portion. Participants explore calculations related to the center of mass and the conditions under which the top may topple when placed on a horizontal surface.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant proposes a calculation for the center of mass, suggesting that it is at y = 0 by symmetry and provides an equation for x, but notes a discrepancy with the textbook answer.
  • Another participant claims to have arrived at the textbook answer by using normalized length units and outlines their calculations involving integrals of a density function.
  • A question is raised about the choice of the density function f(x), with one participant clarifying that it should be proportional to the cross-sectional area and that the specific form is not critical as long as it reflects uniform material properties.
  • Further clarification is provided on the linear density and how it simplifies the problem, with a specific piecewise function for f(x) being discussed.
  • A participant expresses understanding after the explanations, indicating a resolution of their confusion.

Areas of Agreement / Disagreement

There is no consensus on the correct method for calculating the center of mass, as participants have differing approaches and results. Some agree with the textbook answer while others have alternative calculations that yield different results.

Contextual Notes

Participants utilize different assumptions regarding the density function and normalization of units, which may affect their calculations. The discussion includes varying interpretations of the problem setup and the mathematical steps involved.

Who May Find This Useful

This discussion may be useful for students and enthusiasts interested in mechanics, specifically in understanding the concepts of center of mass and stability in physical systems.

vic
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A uniform solid spinning top has the shape of an inverted right circular cone of radius 3r and height 4r surmounted by a right circular of base radius 3r and height 6r. Find the position of the centre of mass of the spinning top and hence show that if it is placed with the curved surface of the cone on a horizontal plane the top will topple.


This is my answer
y = 0 by symmetry

54[Pi]r^3 (3r) + 3[Pi]r^2 (7r) = [54 [Pi] r^3 + 3[Pi] r^2] x

54r^2 + 7r = (18r + 1) x

x = (54 r^2 + 7r) /(18r +1)

But the textbook says the answer is (75/33)r from joint face :confused:


Please Help me. :bugeye:

 
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I got the same answer as the book. To make it easy I used normalized length units with one of my units equalling "4r" of their units. Here's a really quick rough outline of my calculations.

x_c = integral(x f(x) dx) / integral(f(x) dx), where f(x) is the density function, normalized to any level you want.

integral(x f(x) dx) = integral(x^3, x=0..1) + integral(x,x=1..2.5) = 23/8

integral( f(x) dx) = integral(x^2, x=0..1) + integral(1, x=1..2,5) = 11/6

So x_c = 23*6/(11*8) in my normalized units.

In the original units that x_c = (69/11)r from the cones apex which is (25/11)r from the join.
 
Last edited:
How do you know that f(x) is x^2 +1 ??
 
vic said:
How do you know that f(x) is x^2 +1 ??

It's not, but all you need is something that is propotional to the true density since the center of mass will be the same whether it's made of lead or plastic, so long as it's the same uniform material throughout. Actually it's only the linear density I'm considering here, that is mass per unit length, that's all you need consider due to the symmetry in the other two dimensions.

So the linear density, "f(x)", is just a function proportional to the cross-sectional area at the position "x". That's all you need to do and it makes the problem fairly easy.

I made it even easier by choosing length unit of 1 unit = 4r of the original length units. This makes the width of the cylindrical portion be exactly 1 unit and the width of the conic portion changing linearly from 0 units to one unit width over a length of exactly one unit.

So in terms of my length units the cross-section of the conic part is proportional to x^2 and the cross-section of the cylindrical section is proportional to unity (with the same constant of proportionality in each case).

So actually the function I used for f(x) was :

f(x) = x^2 : for 0 <= x <= 1
f(x) = 1 : for 1 < x <= 2.5
f(x) = 0 : otherwise
 
Last edited:
I got it now. Thanks a lot. :)
 

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