A thin rod of length `4l` and mass `4m` is bent at the points as shown in Fig.
What is the moment of inertia of the rod about the axis passing through point `O` and perpendicular to the plane of the paper.

Options

A. `(m l^(2))/(3)`
B. `(10m l^(2))/(3)`
C. `(m l^(2))/(12)`
D. `(m l^(2))/(24)`

Answer

B

Explanation

A thin rod ……………..
The given structure can be broken into 4 parts

For `AB : I = I_(CM)+mxxd^(2)=(ml^(2))/(12)+(5m)/(4)l^(2)`
`, I_(AB)=(4)/(3)ml^(2)`
For `BO : I = (m l^(2))/(3)`
`:.` For composite frame : (by symmetry)
`I = 2[I_(AB)+I_(OB)]= 2[(4m l^(2))/(3)+(m l^(2))/(3)] = (10)/(3)m l^(2).`]