Find out Moment of Inertia of circular area:
Find out Moment of Inertia of circular area of radius a = 10 cm about its centroidal axis OX as illustrated in Figure
Let a thin strip of width By at distance y from axis OX where,
y = a sin θ and By = 2 (a cos θ)
∴ dy = a cos θ d θ
dA = By dy = (2 a cos θ) a cos θ d θ
All elements ought to be considered for values of θ ranging from - 90o to + 90o + 90o
cos 2 θ = cos2 θ - sin 2 θ = 2 cos2 θ - 1
∴ 2 cos2 θ = 1 + cos 2 θ
cos 4 θ = 1 - 2 (sin 2 θ)2
∴ 2 (sin 2 θ)2 = 1 - cos 4 θ
∴ Ix = a4/2 ∫ 2 [ sin 2 θ]2 d θ = a4/2 ∫ (1 - cos 4 θ) d θ
= π a 4 / 4
= π D 4/64 , where D = diameter of circle = 2 a
∴ I x = π× 10 4 /4 = 7854 cm4 .
This is to be noted down that circular cross-section is axis-symmetric, that means it is symmetric w.r.t. both x and y axes or any other radial direction and also has similar nature of shape regarding all centroidal axes.
∴ Iy = Ix = πa4 /4= π a2 (a2/4) = A (a2/4)
where, A = Area of the circle of radius a.
Letting perpendicular axis theorem, we have following :
I z = I x + I y = A (a2/2)