The orem of parallel axis states that if moment of inertia of a plane area about an axis in plane of area ough C.G. of plane area be represented by IG, n moment of inertia of given plane area about a parallel axis AB in plane of area at a distance h from C.G. is given by formula
I_AB = Ah² + I_G.

The centre of gravity of this rectangular area will be half of 3cm from x-axis and half of 12 from y-axis. refore center of gravity will be at (6,1.5).

The moment of inertia of a rectangular section about an horizontal axis through C.G is bd³/12.

Concept:

Area moment of inertia is given by, I = A × k2

Where A is area of section and k is radius of gyration of section.

For circular section, k = D/4

$\mathrm{A}=\frac{\mathrm{Ï€}}{4}{\mathrm{D}}^2$

$\mathrm{k}=\frac{\mathrm{D}}{4}$

$∴\mathrm{I}=\mathrm{A}×{\mathrm{k}}^2=\frac{\mathrm{π}}{64}{\mathrm{D}}^4$

Calculation:

D = 20 cm

Moment of Inertia for a circular section

$∴\mathrm{I}=\frac{\mathrm{Ï€}}{64}{(20)}^4$ = 7854 cm⁴ 

Important Points

Polar Moment of Inertia (J) = 2I = $\frac{\mathrm{Ï€}}{32}{\mathrm{D}}^4$

The product of inertia = area x points of C.G
= ( 10×5) x 5 x 2.5 = 625mm⁴.

The centre of gravity of this rectangular area will be half of 10cm from x-axis and half of 20cm from y-axis. refore center of gravity will be at (10,5).

Area of triangle = 50, area of rectangle = 100
The center of gravity is given by, y = (a₁y₁ + a₂y₂) / (a₁ + a₂) = (50×20/3 + 100×10) / (50+100) = 8.88cm.
This will on for y-axis.
For x-axis, The center of gravity is given by, x = (a₁x₁ + a₂x₂) / (a₁ + a₂) = (50×6.66 + 100×2.5) / (50+100) = 3.88cm.
So center of gravity will be at (2.33, 4.33).

Theorem of perpendicular axis stares that if I_XX and I_YY be moment of inertia of a plane section about two mutually perpendicular axis X-X and Y-Y in plane of section n moment of inertia of section I_ZZ about axis Z-Z, perpendicular to plane and passing through intersection of X-X and Y-Y is given by formula
I_zz ? I_xx = I_yy.

The product of mass and square of distance of center of gravity of mass from an axis is known as mass moment of inertia about that axis.

The center of gravity is given by, y = (a₁y₁ + a₂y₂) / (a₁ + a₂) = (100×17.5 + 150×7.5) / (100+150) = 11.5cm.