What is the Correct Approach to Solving a Challenging Pulley Problem?

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jalen
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please refer to attachment for question =)
 

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Well for one thing, your net force for m2 is wrong. There are two forces acting in the horizontal direction, and two in the vertical. You've mixed them together. You need to write separate equations for each direction.

Where are your equations for m3?

Draw a free body diagram and label the forces. Be careful with your signs. Write out the sum of the forces according to Newton's law for each mass. This will give you a system of equations to work with to find the information you are being asked to solve for.

See if that helps.
 
In your solution for the acceleration, you treat the different tensions as if they were equal. (You also wrote m2 when you meant m3.)