Center of mass of half square without a half circle

SergioQE

1. The problem statement, all variables and given/known data
Find the position of the center of mass for a thin sheet and homogeneous, with sides R and 2R ,from which has been subtracted a half circle of radius R.


[Xcm=(2/3)*R*(4-pi)]


2. Relevant equations

Rcm=(1/M)*∫rdm

3. The attempt at a solution

By simmetry we know Ycm=0.
For de calculus of Xcm we rewrite the equation: Xcm=(1/Area)*∫xdA

dA=dx*y; \[y=2\,R-2\,{\left( {R}^{2}-{x}^{2}\right) }^{0.5}\]
For determinate the dA we take a
Xcm=(1/A)*[itex]\int\2\,x\,R-2\,x\,{\left( {R}^{2}-{x}^{2}\right) }^{0.5}\[/itex]
-R≤x≤R

\[Xcm=\frac{{R}^{3}}{3\,A}\]

\[A=\frac{4\,{R}^{2}-\pi\,{R}^{2}}{2}\]

\[Xcm=\frac{6\,R}{4-\pi}\]

I´ve been attempting in a lot of ways to solve the problem but i didn't get the result. This is an exercise from my exams in Physics I of 1º grade chemical engineering .


PS: I don't know how to make the formulas visible in LaTex

azizlwl

The CM of rectangle R×2R=CM half circle + CM shaded area.

takudo_1912

You should use the guyndanh formula for sheet

Infinitum

I wonder what that is....

SergioQE

Thank you, but when I try doesn't match with the correct answer.

Xcmrectangle= R/2
Xcmcircle= (4/(3*pi))*R

Xcm=(R/2)-(4/(3*pi))*R=0.0755*R

Correct answer: Xcm=(2/3)*R*(4-pi)=0.055*R

SergioQE

I don't know what is that, but i´m pretty sure that the problem can be solved just with the center of mass definition.

Infinitum

That formula is the distance from the base(diameter) of the semi circle... And the semicircle is 'inverted', here

andrien

you should also consider the mass as proportional to area and not just subtract the x's.

azizlwl

That from top.
Require reference to x-axis

SergioQE

Okey, so the center of mass for the semicircle should be R-(4/(3*pi))*R?

Asr=Area of semirectangle, Asc=Area of semicircle.

Then: Xcm=[Asr*(R/2)-Asc*( R-(4/(3*pi))*R)]/(Asr-Asc);

Xcm=0.223*R;(I've reviewed my calculations)

andrien

Xcm=∫(R/2+√(R^2-X^2)/2)(R-√R^2-X^2)dX/∫(R-√R^2-X^2)dX
limits are from 0 to R OR from -R to R that does not matter because of symmetry.It gives that.verify this yourself.

SergioQE

Okay so the dA= (R-√R^2-X^2)dx, I understand that. But why you take x=(R/2+√(R^2-X^2)/2)?
And I tried to solve your equation with Maxima but i didn't get the result.

Yukoel

Hello SergioQE,
You can handle your question without appealing to calculus, provided you know :
(1) Center of mass position of semicircle (removed part) and center mass of full rectangle.
(2) The basic expression for center of mass.
Now the semicircle and the truncated mass (shown figure) form such a system whose center of mass lies on the position of full figure.(use a suitable coordinate system).Please note that they have different masses so you need to be careful.As a matter of fact any calculus derivation of your problem employs the same method.
Does this help?

regards
Yukoel

SergioQE

So why this expression is wrong?
"Asr=Area of semirectangle, Asc=Area of semicircle.

Then: Xcm=[Asr*(R/2)-Asc*( R-(4/(3*pi))*R)]/(Asr-Asc);"

I've considered the different masses, taking lambda= Mass(sr)/Area(sr), and the same for the sc.

And the result still wrong. I've used the coordinate system shown in the figure.

Thanks for answer.

PS: I add the final expression in LaTex for comfort :
[itex]Xcm=[\frac{{R}^{3}-\frac{\pi\,{R}^{2}\,\left( R-\frac{4\,R}{3\,\pi}\right) }{2}}{2\,{R}^{2}-\frac{\pi\,{R}^{2}}{2}}][/itex]

Yukoel

Hello SergioQE
Your expression sounds correct to me and so do your calculations.Your answer seems correct too.
regards
Yukoel

andrien

what do you mean by maxima,just evaluate the integral.in the upper integral,after opening the brackets you will have x^2/2 left whose integral is R^3/6(for 0 to R).the bottom integral is R^2-(∏R^2/4).so center of mass =(R/6)/(1-(∏/4))=4R/6*(4-∏)or 2R/3(4-∏) WHICH IS THE ANSWER YOU GOT THAT.it is now up to you to verify it.
edit:4-∏ is in denominator.

Yukoel

Hello andrien ,
You have used the inverted coordinate system I think(x=0 at the center of the semicircle).Your answer is correct with respect to that.SergioQE's value of x(cm)
and your value add up to R if my calculations are correct .This shows that you have chosen the center as reference for x=0.Correct me if I am wrong.

regards
Yukoel