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Two bodies with masses m1 and m2(m1gtm2)...

Two bodies with masses `m_1` and `m_2(m_1gtm_2)` are joined by a string passing over fixed pulley. Assume masses of the pulley and thread negligible. Then the acceleration of the centre of mass of the system `(m_1+m_2)` is

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Clearly `(m_1 + m_2) gt (m_3 + m_4)` .
Let `T_1` be the tension in left spring and `T_2` that in right spring and T is tension in the string passing over the pulley .
Then for equilibrium of system

`T_1 =m_2g , T=(m_1 + m_2)g`...........(2)
and `m_3g + T_2=T`
From last equation
`T_2 =T-m_3g=(m_1 + m_2-m_3)g` ........ (3)
After the lower thread is cut off
Equation of motion of mass `m_1` is
`m_1g + T_1-T = m_2a_3` .......... (4)
Equation of motion of mass `m_2` is
`m_2g -T_1 = m_2a_2` ...........(5)
Equation of motion mass ` m_3` is
`m_3g + T_2 -T = m_1 a_1`............. (6)
Equation of motion of mass `m_4` is
`m_4g -T_2 =m_4a_4` ............. (7)
Using (1) , (2) and (3) we get
` a_1 = a_2 = a_3 =0`
` and a_4 = (m_4g -T_2)/( m_4) =(m_4g(m_1 + m_2 - m_3)g)/( m_4) = ((m_3 + m_4 - m_1 - m_2)g)/(m_4)`
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