Torque Balance in Rotational Equilibrium

Steps:

• The uniform rod pivoted at its center is in equilibrium, so net torque about the pivot is zero.
• Torque from X: $\tau_X = m_X g d_X$, where $d_X$ is distance from pivot.
• Torque from Y: $\tau_Y = m_Y g d_Y$, acting in opposite direction.
• Set $m_X d_X = m_Y d_Y$, so (from diagram positions).

Why B is correct:

• Balances torques per the principle of moments, with $d_Y = 3 d_X$, yielding $m_X = 3 m_Y$.

Why the others are wrong:

• A: Implies zero mass for X, violating torque balance unless $d_X = 0$, but objects are suspended.
• C: Ratio 2.5 mismatches the 3:1 distance ratio from positions.
• D: Equal masses require equal distances, but positions differ.

Final answer: B