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First Area Moment and Centroidal Co-ordinates

  1. Aug 8, 2007 #1

    app

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    1.The problem statement, all variables and given/known data
    There is a quarter circle.The centre of the circle of which the quarter circle is a part,is at the origin (0,0).Find the centroidal co-ordinates for the quarter circle.The equation of the circle for which the quarter circle is a part,is x2 +y2 = R2. where R is radius.

    2.Relevant equations
    M(y)= integral of (x dA), where dA is an elemental area at a distance x from Y axis.M(y) is first moment of inertia about y axis. X(c)= M(y)/A, where A is total area of quarter circle,and X(c) is the centroidal x co-ordinate.

    3.The attempt at a solution
    Well,i am in engineering first year and we have just started first area moment,centrodal co-ordinates,second area moment etc.So,i tried my best to understand the concepts.I did one where there is a rectangle with height h and base b.Thats easy.If we first choose a vertical strip of elementary area dA and width dx.Now,M(y)= integral of (x dA).Putting h dx in place of dA,we get an integrable equation.The limits of intigration is from 0 to b.Then,X(c)= M(y)/A.Putting the value of M(y) and A=bh,we get X(c).Similarly by choosing a horizontal strip we can get M(x),which is the first area moment about x axis.I tried to do the same for the quarter circle but I am not being able to get an integrable form.

    4.Conclusion
    Well,I have tried to show that I have attempted the problem and I do study.But since I'm not a genius,I need your help.Please explain the problem,because I may be a bit dumb,although I got very good marks in my school final exams.That time also,i had posted some questions on physics forums.Thanks a lot for everything...
     
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  3. Aug 8, 2007 #2

    Dick

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    Start by expressing M(y) (your integral x*dA) as an explicit integral. It will be a double integral over x and y. What will be appropriate limits of integration?
     
  4. Aug 8, 2007 #3

    app

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    I'm sorry,but that is what i'm not being able to do.could u please explain it?The appropraite limits of integration will be from 0 to r,where r is radius of quarter circle.
     
  5. Aug 8, 2007 #4

    Dick

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    Ok, let's integrate dx from 0 to R. Inside of that integral we have to integrate dy. To get the y range, picture a thin vertical rectangle at some value of x, stretching from the x-axis to the quarter circle (since that's the interior of the region of integration). Got it? So now if I said the y limits are from 0 (the x axis) to sqrt(R^2-x^2) (the circle), would you believe me?
     
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