The centroid of volume is point where total volume is assumed to be concentrated. It is geometric centre of a body. If density is uniform throughout body, n center of mass and center of gravity correspond to centroid of volume. The definition of centroid of volume is written in terms of ratios of integrals over volume of body.

The product of inertia = area x C.G
= ?x 10×10 / 4 x 5×5 = 1963mm².

Area of triangle = 25, area of rectangle = 100
The center of gravity is given by, y = (a₁y₁ + a₂y₂) / (a₁ + a₂) = (25×10/3 + 100×5) / (25+100) = 4.66cm.
This will on for y-axis.
For x-axis, The center of gravity is given by, x = (a₁x₁ + a₂x₂) / (a₁ + a₂) = (25×11.66 + 100×5) / (25+100) = 6.332cm.
So center of gravity will be at (2.33, 4.33).

Centre of gravity of various plane areas

1) Triangle

2) Semi circle of radius ‘R’

3) This Hollow Cone (Right angled)

4) Trape zoid

$\stackrel{¯}{y}=\frac{ga+6}{\left(a+6\right)}\left(\frac{4}{3}\right)$

5) sine wave

6) 4^th degree curve

$\stackrel{¯}{x}=\left(\frac{6\left(N+1\right)}{2\left(N+2\right)}\right)$

$\stackrel{¯}{y}=\left(\frac{hN}{ZN+1}\right)$

Important Points

1. Centre of gravity of a thin hollow cone = 1/3
2. Centre of gravity of a solid cone = 1/4

The moment of inertia of a triangular section about base is bh³/12
The moment of inertia of a triangular section about an axis passing through C.G. is bh³/36
So difference = bh³/12 ? bh³/36 = bh³/18.

The centroid is point at which total area of a plane figure is assumed to be concentrated. The centroid and centre of gravity are at same point.

The moment of inertia of a triangular section about base is bh³/12.

Area of upper web a1 = 16cm, area of flange a2 = 24, area of lower web a3 = 32.
The center of gravity is given by, y = (a₁y₁ + a₂y₂ + a₃y₃) / (a₁ + a₂ + a₃) = (16×15 + 24×8 +32×1 / (16 +24+32)) = 6.44cm.

The whole weight of a circle can be assumed to act through its center. So center of gravity of a circle is at its center.

The moment of inertia of a circular section is ?D⁴/64.