Find out moment of inertia of circular area, Mechanical Engineering

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

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Solution

 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

1760_Find out Moment of Inertia of circular area1.png

All elements ought to be considered for values of θ ranging from - 90o to + 90o + 90o

1468_Find out Moment of Inertia of circular area2.png

Now that

cos 2 θ = cos2 θ - sin 2 θ = 2 cos2 θ - 1

∴ 2 cos2 θ = 1 + cos 2 θ

and,

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.

1046_Find out Moment of Inertia of circular area3.png

∴ 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)

Posted Date: 1/29/2013 1:26:41 AM | Location : United States







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