Refer to the edited figure shown below.
Pulleys A, B, and C are considered massless and they are stationary. The ropes are considered massless.
Because the pulleys do not move, net tensions should be zero on each pulley.
Therefore the tensions in the ropes are as shown.
Pulley A experiences two downward T₁ tensions that are balanced by one upward tension of 2T₁.
Pulley B experiences two downward tensions of 2T₁ that are balanced by one upward tension of 4T₁.
Pulley C experiences two upward tensions of T₁ and 2T₁ that are balanced by a downward tension of 3T₁.
The top support does not move. Therefore the top tension should balance the total weight of the two masses. That is,
4T₁ = m₁g + m₂g = (m₁ + m₂)g (1)
Let a₁ and a₂ be the accelerations of masses m₁ and m₂ respectively.
The motion of m₁ obeys the equation
m₁a₁ = m₁g - T₁
= m₁g - (1/4)(m₁ + m₂)g
m₁a₁ = (g/4) (3m₁ - m₂)
a₁ = (g/4)[3 - m₂/m₁] (2)
The motion of m₂ obeys the equation
m₂a₂ = m₂g - 3T₁
= m₂g - (3/4)(m₁ +m₂)g
= (g/4)(m₂ - 3m₁)
a₂ = (g/4)[1 - 3(m₁/m₂)]
Answer:
The accelerations of m₁ and m₂ are respectively
[tex]a_{1} = \frac{g}{4} (3- \frac{m_{2}}{m_{1}}) \\ \\ a_{2} = \frac{g}{4}(1-3 \frac{m_{1}}{m_{2}}) [/tex]
