For this pulley system, why is it m1g - T1-T2=m1a?

[paki123]

Three objects with masses m1 = 66.0 kg, m2 = 37.4 kg, and m3 = 21.8 kg are hanging from ropes that run over pulleys. What is the acceleration of m1? (Take the upward direction to be positive.)

So for the two outside blocks,

I know that the forces acting on the blocks are T1=m2g+m2a, and T2=m3g+m2a

But why in literally every break down of this problem I see, the force acting on the main block is m1g - T1-T2 = m1a, but not T1+T2-m1g=m1a?

I can't seem to get the answer when I do it with the latter force layout because I end up with a negative not belonging somewhere.

 

[Nugatory]

I know that the forces acting on the blocks are T1=m2g+m2a, and T2=m3g+m2a

But why in literally every break down of this problem I see, the force acting on the main block is m1g - T1-T2 = m1a, but not T1+T2-m1g=m1a?

I think you have a typo in the T2 equation, which I've marked in boldface.

The difference between m1g - T1-T2 = m1a and T1+T2-m1g=m1a is just the difference between choosing the positive direction up and the positive direction down; the two left-hand sides are just negatives of one another. Be sure you're consistent in applying your positive-up convention, and consider (no math required for this part, just common sense) whether for this particular combination of weights you expect the middle mass to be accelerating upwards or downwards. Whatever the middle mass does, the end masses will have the opposite sign because they're going in the other direction.

 

[Puneeth423]

Three objects with masses m1 = 66.0 kg, m2 = 37.4 kg, and m3 = 21.8 kg are hanging from ropes that run over pulleys. What is the acceleration of m1? (Take the upward direction to be positive.)

So for the two outside blocks,

I know that the forces acting on the blocks are T1=m2g+m2a, and T2=m3g+m2a

But why in literally every break down of this problem I see, the force acting on the main block is m1g - T1-T2 = m1a, but not T1+T2-m1g=m1a?

I can't seem to get the answer when I do it with the latter force layout because I end up with a negative not belonging somewhere.

When ever you do pulley problems, follow simple basics(drawing a free body diagram) and laws. If you over think, things become more complex. Here in the given case you don't know in what direction m1/m2/m3 moves, since it is nowhere given in the question that m1 moves down and m2 and m3 move up. You have taken it for granted that m2 and m3 move up which makes m1 move down. The question i ask you here is what made you think that m1 moves down?