The centre of mass of circle section.

Norbertto

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Jun 18, 2019
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Hi !
I'm new user of that forum and I have a problem with centre of mass in circle section.
What is the distance between main centre of mass of whole circle and the centre of mass of circle section.
Unfortunatelly I can't find suitable formula.
Look at the photo I have attached.
 

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tkhunny

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Can you find the Center of Mass of the WHOLE sector?
How about the Center of Mass of the isosceles triangle alone?
Subtraction would work after those two are found.
 

HallsofIvy

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Take the center of the circle to be the origin of a coordinate system with the y-axis the perpendicular bisector of the segment. Then the equation of the circle is \(\displaystyle x^2+ y^2= R^2\). With angle \(\displaystyle \alpha\), hypotenuse R, the distance from the center of the segment to the end is \(\displaystyle R sin(\alpha)\) while the distance from the center of the circle to the segment is \(\displaystyle R cos(\alpha)\). That is, the equation of the line is \(\displaystyle y= R cos(\alpha)\) (a constant) and its two end points are \(\displaystyle (-Rsin(\alpha), Rcos(\alpha))\) and \(\displaystyle (Rsin(\alpha), Rcos(\alpha)\).
The area of that region is \(\displaystyle \int_{-Rsin(\alpha)}^{Rsin(\alpha)} (\sqrt{R^2- x^2}- R) dx\).
The x component of the centroid (not center of mass- there is no mass) is, by symmetry, x= 0. To find the y component of the centroid find \(\displaystyle \int_{-Rsin(\alpha)}^{Rsin(\alpha)}(R^2- x^2 - R\sqrt{R^2- x^2})dx\) and divide by the area.
 
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Dr.Peterson

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Nov 12, 2017
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Hi !
I'm new user of that forum and I have a problem with centre of mass in circle section.
What is the distance between main centre of mass of whole circle and the centre of mass of circle section.
Unfortunatelly I can't find suitable formula.
Look at the photo I have attached.
The region you're asking about is called a segment of a circle. If all you need is a formula, you could search for "center of mass of segment of a circle" (or "centroid"). The link includes such a formula, which is not pretty.

If you need to be able to figure it out for yourself, probably using calculus, you'll need to show us where you are stuck doing the integration.

As tkhunny suggests, if you have formulas for the center of mass of a sector and of a triangle, you could create your formula by subtraction -- but it would have to be a weighted average, taking the area/mass of each part into account. See here or here, for the addition formula.
 
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