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(B) Let the mass $m_1 = m$ and $m_2 = 3m$. The acceleration of the masses in an Atwood machine is given by $a = \frac{m_2 - m_1}{m_1 + m_2} g = \frac{

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$-g/4$

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(B) Let the mass $m_1 = m$ and $m_2 = 3m$. The acceleration of the masses in an Atwood machine is given by $a = \frac{m_2 - m_1}{m_1 + m_2} g = \frac{3m - m}{3m + m} g = \frac{2m}{4m} g = \frac{g}{2}$.Taking the downward direction as positive for $3m$ and upward for $m$,the acceleration vectors are $\vec{a}_1 = \frac{g}{2} \hat{j}$ and $\vec{a}_2 = -\frac{g}{2} \hat{j}$.The acceleration of the center of mass is given by $\vec{a}_{cm} = \frac{m_1 \vec{a}_1 + m_2 \vec{a}_2}{m_1 + m_2}$.Substituting the values: $\vec{a}_{cm} = \frac{m(\frac{g}{2}) + 3m(-\frac{g}{2})}{m + 3m} = \frac{\frac{mg}{2} - \frac{3mg}{2}}{4m} = \frac{-mg}{4m} = -\frac{g}{4}$.

$A$ pulley fixed to the ceiling carries a string with blocks of mass $m$ and $3m$ attached to its ends. The masses of the string and pulley are negligible. When the system is released,what is the acceleration of its center of mass?

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