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Centroid, the y-direction centroid distance of the area surrounded by y = ax^n and the y-axis

작성자 Uploader : pleiades 작성일 Upload Date: 2019-03-07변경일 Update Date: 2019-10-22조회수 View : 367

As shown in the figure, the y-direction centroid distance Gy of the area surrounded by y = ax^n and the y-axis can be obtained as follows. 
 
Gy = Mx / A

Mx : first moment of inertia relarive to y axis
A : area

To simplify, calculating the area with respect to y = y1, then subtracting it from y1.

My1 = ∫(y1-y)*(y1-y)/2 dx = (a/2)∫(x1^n - x^n)*(x1^n - x^n) dx

    = (a/2)*∫x1^(2n) - 2*x1^n*x^n + x^(2n) dx

    = (a/2)*(x1^(2n)*x - (2/(n+1))*x1^n*x^(n+1) + (1/(2n+1))*x^(2n+1))

Since x = x1,

My1 = (a/2)*(x1^(2n+1) - (2/(n+1))*x1^(2n+1) + (1/(2n+1))*x1^(2n+1))

    = (a/2)*(x1^(2n+1))*(1 - 2/(n+1) + 1/(2n+1))

    = (a/2)*(x1^(2n+1))*((n+1)*(2n+1) - 2*(2n+1) + (n+1))/((n+1)*(2n+1))

    = (a/2)*(x1^(2n+1))*(2n^2)/((n+1)*(2n+1))
   
    = a*(x1^(2n+1))*(n^2)/((n+1)*(2n+1))

Firstly, calculating the area to x-axis,

Ax = ∫ y dx = ∫ a*x^n dx = (a/(n+1))*x^(n+1)

Then, the area from 0 to x1 is :

Ax = (a/(n+1))*x1^(n+1)

Subtracting this from x1*y1,

A = x1*y1 - Ax = x1*y1 - (a/(n+1))*x1^(n+1)

Since y1 = ax1^n,

A = ax1^(n+1) - (a/(n+1))*x1^(n+1)

 = a*(1-1/(n+1))*x1^(n+1)

 = a*(n/(n+1))*x1^(n+1)

Gy1 = (a*(x1^(2n+1))*(n^2)/((n+1)*(2n+1)) / (a*(n/(n+1))*x1^(n+1))

    = ((x1^(n))*(n^2)/((n+1)*(2n+1)) / ((n/(n+1)))

    = ((x1^(n))*(n)/((2n+1))
 
Gy = y1-Gy1 = ax1^n - ((x1^(n))*(n)/((2n+1))

  = ((a - n/(2n+1))*x1^n

*** 참고문헌[References] ***
Gy = (a - n/(2*n+1))*x1^n
변수명 Variable 변수값 Value 변 수 설 명 Description of the variable


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