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Equation for uniform acceleration in Double Atwood

Double Atwood machine.

We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inelastic string and an ideal massless pulley the only forces we can consider are the two tension forces ($''T_{1}'',''T_{2}''$), and the weights of the three masses ($''m_{A}'',''m_{B}'',''m_{C}''$). To find the accelerations we need to consider the accelerations and the forces affecting each individual mass.

We know that:

$T_{1}-m_{A}*g=m_{A}*a_{A}$
$T_{1}-m_{B}*g=m_{B}*a_{B}$