Centroid of y = kx² area; parallel axes; define centroid

Centroid of the plane figure

Reading of the figure: the area lies under the curve y = kx² from x = 0 to x = a, where the height is h, so k = h/a². Strip: width dx, height y = kx².

Area: A = ∫₀ᵃ kx² dx = k a³/3 = ah/3

x̄: ∫ x dA = ∫₀ᵃ k x³ dx = k a⁴/4 = a²h/4 x̄ = (a²h/4)/(ah/3) = 3a/4

ȳ: ∫ ȳ dA = ∫₀ᵃ ½ k² x⁴ dx = k² a⁵/10 = a h²/10 ȳ = (a h²/10)/(ah/3) = 3h/10

Centroid = (3a/4, 3h/10) from O.

Parallel axis theorem

Statement: the moment of inertia of an area about any axis equals its moment of inertia about a parallel axis through its centroid plus the product of the area and the square of the distance between the two axes.

I = I_G + A d²

Proof: let G be the centroid of the area A, and let X′ be the centroidal axis. Take a parallel axis X at distance d from X′. For a small element dA at distance y′ from X′, its distance from X is y = y′ + d.

I_X = ∫ y² dA = ∫ (y′ + d)² dA
    = ∫ y′² dA + 2d ∫ y′ dA + d² ∫ dA

• ∫ y′² dA = I_G, the moment of inertia about the centroidal axis.
• ∫ y′ dA = A·ȳ′ = 0, because the distance of the centroid from the centroidal axis is zero.
• ∫ dA = A.

So I_X = I_G + A d². Similarly, I_Y = I_Gy + A d_x². Since A d² is never negative, the moment of inertia about the centroidal axis is the least of the moments of inertia about all parallel axes.

Centroid and centre of gravity

• Centroid: the geometric centre of an area (or line or volume). It is the point at which the whole area can be assumed concentrated; x̄ = ∫x dA/A, ȳ = ∫y dA/A. It depends only on shape.
• Centre of gravity: the point through which the resultant of the weights of all the particles of a body acts, whatever the orientation of the body. It depends on the weight distribution. For a body of uniform density the centre of gravity and the centroid coincide.