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(D) The moment of inertia of the system about axis $AA$ (passing through $m_A$) is $I_{AA} = m_B l^2$.The moment of inertia of the system about axis $

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$(1/4) l$

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(D) The moment of inertia of the system about axis $AA$ (passing through $m_A$) is $I_{AA} = m_B l^2$.The moment of inertia of the system about axis $BB$ (passing through $m_B$) is $I_{BB} = m_A l^2$.Given that $\frac{I_{BB}}{I_{AA}} = 3$,we have $\frac{m_A l^2}{m_B l^2} = 3$,which implies $\frac{m_A}{m_B} = 3$ or $m_A = 3 m_B$.Let $x_A$ be the distance of the centre of mass from mass $m_A$. Then the distance from $m_B$ is $(l - x_A)$.By the definition of the centre of mass,$m_A x_A = m_B (l - x_A)$.Substituting $m_A = 3 m_B$,we get $3 m_B x_A = m_B (l - x_A)$.Dividing by $m_B$,we get $3 x_A = l - x_A$,which simplifies to $4 x_A = l$.Therefore,$x_A = \frac{l}{4}$.

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