We are now in position to find the accelerations a1, a2,
and the tension, because we have Newton's second law
and our constraint condition for the acceleration.
Let's recall the equations that we found.
We had m1g minus t was m1a1.
And we had m2g minus 2t was equal to m2a2.
And we also had the condition-- constraint condition
between the accelerations that a1 was minus 2a2.
So now we have a system of three equations.
And we can find-- we can solve for any of these quantities
that we want.
So one way to do it is to identify an equation
and to identify a quantity that we would like to solve for.
So for instance, let's identify that we
want to solve for a1 first.
So how are we going to develop a strategy for that?
Well, I'll choose equation 1 as my backbone equation.
And I have an unknown t.
But in equation 2, that unknown t is appearing there,
but it's expressed in terms of a2.
And a2 , though, from equation 3 is expressed in terms of a1.
So my first step is to write equation-- rewrite equation
2 as m2g minus 2t equals m2.
Now here I'm going to make the substitution, which
is minus a1 over 2, so we'll call that 2a.
And now I have equation 1 and equation 2a,
two unknowns t, and a1.
And what I can do is I can solve for either equation.
And I can make a choice what's easiest.
When I look at these equations, it's
easiest for me to identify what t is in terms of a1.
So I'll write m1g minus m1a1 is equal to t.
I'll call that equation 1a.
And now I can substitute that value of t into equation 2a.
And I get m2g minus 2 times m1g minus m1a1.
And that's equal to minus m2 over 2 times a1.
And now I'd like to collect my a1 terms.
And what I have over here-- let's
bring all the a1 terms to this side.
And I get m2g minus 2m1g.
I have a plus, so I'm going to bring that over to this side.
And I get equal to-- that's plus plus,
so I'll have two minus signs, and I'll
have 2m1 plus m2 over 2 a1.
And now I can solve for a1.
And what I get is I get 2m1g minus m2g.
And I want to divide through by this denominator.
And what I have downstairs is 2m1 plus m2 divided by 2.
And that's my expression for a1.
Once I have that expression for a1,
I can easily come back and find out what a2 is.
Or I can substitute it into the equation here for a1
and find out what the tension is.
And so I can now easily find my expressions for a2 and t.
But I'll leave that as an exercise.