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Recognize Tinitial=5mg

 

I started with free body diagrams of both masses, and realizing the tension is the same, putting the two Newton's 2nd Law equations together to solve for the acceleration.  T-4mg=4ma, 5mg-T=5ma.  Solve for a=g/9

 

Next, I put the acceleration back into the Newton's 2nd Law equations to find Tfinal=40mg/9

 

Then, you know that Ti-Tf=5mg-40mg/9=X N, solve for m.

 

Once you have that, part b and c get pretty straightforward.

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