Geometric expressions for a quarter circle cut at an arbitrary point

  • #1

Homework Statement



I am after finding general geometric expressions for a quarter-circle that is split into two segments along either its domain or range (they are equal). I.e. Taking the circle shown in Figure 1 and concentrating on the upper right quadrant, I am after expressions for the individual areas (top and bottom) and their respective centroids x1, y1 and x2, y2 (Figure 2) when cut at "c".

Figure 1:
AL6SzPl.jpg


Figure 2:
HZgQwgo.jpg


Homework Equations



Area of circle = ∏r[itex]^{2}[/itex]

Equation of quarter circle: y(x) = √(x[itex]^{2}[/itex]-r[itex]^{2}[/itex])


The Attempt at a Solution



Apart from stating the obvious equations (above) I’m stuck on this problem.

I have searched my textbooks/google (using general key words) for expressions of this particular case with no luck.

I have never been strong with deriving expressions from first principals and would appreciate any ideas/push in the right direction. This is a problem directed at self-study, not homework.
 

Answers and Replies

  • #2
If you immagine the other half of the circle you can use the equation:
[itex]\bar{y}[/itex]=[itex]\frac{4Rsin^{3}(\frac{1}{2}θ[STRIKE][/STRIKE])}{3(θ-sin(θ))}[/itex]
where θ is the theoretical angle to create the chord to get your y bar for the top section. Then you can calculate the y bar for the whole quadrant by using the equation:
[itex]\bar{y}[/itex]=[itex]\frac{4R}{3π}[/itex]
Since the average of the top and bottom y bar have to equal the quadrant based on their area you can take the weighted average of the top area and the y bar with the weighted average of the bottom area equaling the total area and the total y bar. From that you can get your y bars and then you can do the same thing for the x bars. Might sound confusing so if you don't understand let me know, I'll try my best.

Area of top part:
A=[itex]\frac{R^{2}}{2}[/itex](θ-sin(θ))
 

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