Four thin uniform rods each of mass m and length I are arranged to form a square. Find the moment of inertia of the system about an axis (i) Passing through its centre and perpendicular to its plane. (ü) Passing through one of its sides. (üi) Passing through a corner and perpendicular to its plane. (iv) About a diagonal of the system

Explanation

(i) M.I. about an axis passing through its centre and perpendicular to its plane :
`4[(ml^(2))/(12)+m((l)/(2))^(2)]=(4ml^(2))/(12)xx4`
(ii) `I_(s)=(ml^(2))/(3)xx2+ml^(2)`
`=(5)/(3)ml^(2)`
(iii) From parallel axes theorem
`I=I_(C )+mr^(2)=(4ml^(2))/(3)+4m((l)/(sqrt(2)))^(2)=(10)/(3)ml^(2)`
(iv) From perpendicular axes theorem
`I_(z)=I_(x)+I_(y) " " I_(z)=2I_(x)`
`I_(x)=(I_(z))/(2)=(I_(C ))/(2)=(2)/(3)ml^(2)`