First Area Moment and Centroidal Co-ordinates

In summary, the conversation discusses finding the centroidal co-ordinates for a quarter circle, with the equation x^2 + y^2 = R^2. The relevant equations are M(y) = integral of (x dA) and X(c) = M(y)/A. The attempt at a solution involves using a double integral over x and y, with appropriate limits of integration from 0 to R. The conclusion is that the person needs help understanding the problem and asks for an explanation.
  • #1
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1.Homework Statement
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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  • #2
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?
 
  • #3
Dick said:
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?

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.
 
  • #4
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?
 

FAQ: First Area Moment and Centroidal Co-ordinates

What is the first area moment?

The first area moment, also known as the first moment of area or simply moment of area, is a mathematical concept used in engineering and physics to calculate the distribution of forces or moments acting on a body. It is calculated by multiplying the area of an object by its distance from a reference axis.

How is the first area moment used in engineering?

In engineering, the first area moment is used to determine the centroid or center of mass of an object. This information is crucial in designing structures and machines, as it helps in understanding how forces and loads are distributed and how the object will behave under different conditions.

What are centroidal co-ordinates?

Centroidal co-ordinates are a set of coordinates that define the location of the centroid of an object. They are typically expressed as (x, y, z) coordinates, where x and y represent the distances from the reference axes and z represents the distance from the plane of the object.

Why is knowing the centroidal co-ordinates important?

Knowing the centroidal co-ordinates of an object is important because it allows engineers to accurately calculate the moments of inertia and other mechanical properties of the object. This information is necessary for designing and analyzing structures and machines.

How can the first area moment and centroidal co-ordinates be calculated?

The first area moment and centroidal co-ordinates can be calculated using mathematical formulas and integrals. The specific equations used depend on the shape and dimensions of the object. There are also computer programs and software that can assist in these calculations.

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