A triangular of rigid wire frame `'AOB'` is made, in which length of each wire is `l` and mass `m`. The whole system is suspended from point `O` and free to perform `SHM` about `x`-axis or about `z-`axis. When it performs `SHM` about `x-`axis its time period of oscillation is `T_(1)` and when it performs `SHM` about `z-`axis, its time-period of osciallation is `T_(2)`, then choose the correct option.

Options

A. `T_(1) lt T_(2)`
B. `T_(1) gt T_(2)`
C. `T_(1) = T_(2)`
D. None

Explanation

`T=2pisqrt((1)/(M_(sys)gd))`, `M_(sys)=3m`, `d=(lsqrt(3))/(2)xx(2)/(3)=(l)/(sqrt(3))`
Case I : System is performing SHM about `x-`axis
`I_(X)=I_(AB)+I_(AO)+I_(BO)=(3ml^(2))/(4)+[(ml^(2))/(3)sin^(2)60^(@)]xx2=(5ml^(2))/(4)`
`T_(1)=2pisqrt((I_(x))/(M_(sys)gd))=2pisqrt(((5ml^(2))/(4)xxsqrt(3))/(3mxxgxxl))=2pisqrt((5sqrt(3)l)/(12g))`
Case II : System is performing SHM about `z-` axis
`I_(z)=I_(AB)+I_(AO)+I_(BO)=[(ml^(2))/(12)+m((sqrt(3)l)/(2))^(2)]+(ml^(2))/(3)+(ml^(2))/(3)=ml^(2)[(1)/(12)+(3)/(4)+(2)/(3)]`
`=ml^(2)[(1+9+8)/(12)]=(3ml^(2))/(2)`
`T_(2)=2pisqrt((I_(z))/(M_(sys)gd))=2pisqrt((3ml^(2)xxsqrt(3))/(2xx3mxxgxxl))=2pisqrt((sqrt(3)l)/(2g))`