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    Arbelos

    Definition

    Defining inequalities

    x^2 + y^2<=a^2/4 and y>=0 and y^2>=(a/2 + x) (1/2 a (2 r - 1) - x) and 1/4 a^2 (2 r - 1) + x^2 + y^2>=a r x

    Alternate name

    shoemaker's knife

    Lamina properties

    a>0 and 0<r<1

    A = 1/4 π a^2 (1 - r) r

    x^_ = (1/2 a (-1/2 + r), a/π)

    Mechanical properties

    J_x invisible comma x = -1/64 π a^4 (r - 1) r ((r - 1) r + 2)

    J_y invisible comma y = -1/64 π a^4 (r - 1) r (5 (r - 1) r + 2)

    J_zz = 1/32 π a^4 r (-3 r^3 + 6 r^2 - 5 r + 2)

    J_x invisible comma y = 1/24 a^4 (r - 1) r (2 r - 1)

    r_x = 1/4 a sqrt((r - 1) r + 2) r_y = 1/4 a sqrt(5 (r - 1) r + 2)

    Distance properties

    p = 2 π a

    a

    Classes

    concave laminae | curvilinear laminae

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