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# Consider the following Atwood machine with three masses in the ratio m$_1$: m$_2$: m$_3$:: 1:2:3, hung with a massless string over a friction-less pulley. What is the tension in the string between masses m$_2$ and m$_3$?

Answer: $mg$
The Free Body diagrams for the problem can be drawn as follows:
Net force in the direction of motion of $m_1$ is $F_1 = T- m_1g =m_1a$ as per Newton's second law.
Net force in the direction of motion of $m_2$ is $F_1 = T- T'=m_2a$ as per Newton's second law.
Net force in the direction of motion of $m_3$ is $F_3 = m_3g- T'=m_3a$ as per Newton's second law.
Solving for acceleration, we get: $a = \large\frac{m_2 + m_3 - m_1}{m_1+m_2+m_3}$$g = \large\frac{2m+3m-m}{m+2m+3m}$$g = \large\frac{2}{3}$$g Now, m_3g -T' = m_3 a \rightarrow T' = m_3 (g-a). Subsituting for a and for m_3 = 3m, we get: T' = 3m (g - \large\frac{2}{3}$$g) = mg$