Why Can't I Solve for a2 Instead of a1 in Connected Objects Problem?

[CVRIV]

I attached an image of the problem and solution from the book and my attempt at solving it. I don't need help solving the problem the way the book explains it; i understand that.

What I don't understand is why I can't solve for (a2) instead of (a1). The book says a2 = -a1, which makes sense to me, because the magnitudes are the same except the polarity for a2 is negative.

What about a1 = -a2? I tried solving the problem the same way except I substituted -a2 for a1 instead of the other way around and it doesn't work. I just don't understand why it wouldn't work.

Please help me understand this.

 

[mfb]

You need to keep the signs consistent. In your first equation a positive a1 means the mass accelerates towards the edge. In your second equation a positive a2 means the mass accelerates downwards. That is possible - but then you have a1=a2 without the minus sign.

 

[CVRIV]

Why would positive a2 accelerate downwards? I thought positive a2 accelerates upwards?

In equation 2 I divided both sides by -1 so that I could cancel out the tension T. Was that not right?

 

[CVRIV]

Oh wow. I think I know what I did wrong. How stupid of me.

 

[mfb]

Why would positive a2 accelerate downwards? I thought positive a2 accelerates upwards?

It is arbitrary which direction you choose, but you have to be consistent.

 

[CVRIV]

Finally! The problem was that from the very start I was inverting the polarity of m1a2. I kept writing down T - m2g = -m2a2. I did that because I had it in my head that a2 was negative, which it is, but only after solving for a2. By assigning it as negative from the start, I was in fact just screw it all up. Also... I had it stuck in my head that I had to add the two equations together. I went back to the previous problem in the book, which as a Atwood's Machine problem, and it was subtracting the equations. I I thought I was doing it wrong so I told myself I have to subtract the problems. That's when I realized, fully realized, that it totally depended on the cancellation of T. I get it now. Thanks for your help.