Finding the center of mass of a piecewise function

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Homework Help Overview

The discussion revolves around finding the center of mass (COM) of a piecewise function, specifically involving shapes that can be divided into parallelograms and triangles. Participants are exploring the use of integrals and geometric methods to determine the COM.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss dividing the shape into simpler geometric forms to calculate the COM. There is mention of using integrals versus calculating COM for individual shapes. Questions arise regarding the implications of integrating equations and how that relates to finding the COM versus area.

Discussion Status

The conversation is active, with participants offering different methods for approaching the problem. Some guidance has been provided regarding the use of point masses and integration, but there remains confusion about the relationship between area and center of mass calculations.

Contextual Notes

Participants are considering the implications of the specific shapes involved and the equations provided for the design of the lightning bolt shape. There is uncertainty about the definitions and assumptions regarding the shapes being analyzed.

JorkThePork
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Homework Statement
Find the center of mass of the "lightning bolt" shaped piecewise function as shown below (or in the desmos project: https://www.desmos.com/calculator/g6crwsecp1)
Relevant Equations
Xcm = A^-1 * ∫ a b x * (f(x) - g(x)) d x

Ycm = A^-1 * ∫ a b .5((f(x)^2 - g(x)^2) d x
desmos-graph.png
I understand that I can divide this shape into a few parallelograms and a triangle and calculate the center of mass of each, but am confused as to what I should do after that. My physics teacher also wants us to use integrals, but I'm assuming I can calculate the COM of each parallelogram and triangle by simply integrating.
 
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You can find the CM of the two parallelograms and the half-parallelogram which will give you 3 point masses at the locations of these CMs. Then you can find the CM of the 3 point masses the usual way.
 
kuruman said:
You can find the CM of the two parallelograms and the half-parallelogram which will give you 3 point masses at the locations of these CMs. Then you can find the CM of the 3 point masses the usual way.
the two seemingly big parallelograms actually are not parallelograms because of how I designed the lightning bolt (the equations I used are here https://www.desmos.com/calculator/g6crwsecp1). However, I could break the shape up into 2 big parallelograms, 2 very thin parallelograms, and a triangle. I'm assuming I could still then find the CM of the 5 point masses the usual way.
 
JorkThePork said:
the two seemingly big parallelograms actually are not parallelograms because of how I designed the lightning bolt (the equations I used are here https://www.desmos.com/calculator/g6crwsecp1). However, I could break the shape up into 2 big parallelograms, 2 very thin parallelograms, and a triangle. I'm assuming I could still then find the CM of the 5 point masses the usual way.
Yes, you can carve it up into simple non overlapping shapes, but since you have equations why not just integrate them and add the (signed) results?
 
haruspex said:
Yes, you can carve it up into simple non overlapping shapes, but since you have equations why not just integrate them and add the (signed) results?
Sorry, I’m a little confused how this would give me the center of mass. wouldn’t this just give the area?
 
JorkThePork said:
Sorry, I’m a little confused how this would give me the center of mass. wouldn’t this just give the area?
So what do you need to multiply the expressions by before integrating?
 
Last edited:

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