WEBVTT

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We are now in position to
find the accelerations a1, a2,

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and the tension, because
we have Newton's second law

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and our constraint condition
for the acceleration.

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Let's recall the
equations that we found.

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We had m1g minus t was m1a1.

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And we had m2g minus
2t was equal to m2a2.

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And we also had the condition--
constraint condition

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between the accelerations
that a1 was minus 2a2.

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So now we have a system
of three equations.

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And we can find-- we can solve
for any of these quantities

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that we want.

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So one way to do it is
to identify an equation

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and to identify a quantity that
we would like to solve for.

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So for instance,
let's identify that we

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want to solve for a1 first.

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So how are we going to
develop a strategy for that?

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Well, I'll choose equation
1 as my backbone equation.

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And I have an unknown t.

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But in equation 2, that
unknown t is appearing there,

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but it's expressed
in terms of a2.

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And a2 , though, from equation
3 is expressed in terms of a1.

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So my first step is to write
equation-- rewrite equation

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2 as m2g minus 2t equals m2.

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Now here I'm going to make
the substitution, which

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is minus a1 over 2,
so we'll call that 2a.

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And now I have equation
1 and equation 2a,

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two unknowns t, and a1.

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And what I can do is I can
solve for either equation.

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And I can make a
choice what's easiest.

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When I look at these
equations, it's

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easiest for me to identify
what t is in terms of a1.

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So I'll write m1g minus
m1a1 is equal to t.

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I'll call that equation 1a.

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And now I can substitute that
value of t into equation 2a.

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And I get m2g minus 2
times m1g minus m1a1.

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And that's equal to
minus m2 over 2 times a1.

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And now I'd like to
collect my a1 terms.

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And what I have
over here-- let's

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bring all the a1
terms to this side.

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And I get m2g minus 2m1g.

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I have a plus, so I'm going to
bring that over to this side.

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And I get equal to--
that's plus plus,

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so I'll have two
minus signs, and I'll

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have 2m1 plus m2 over 2 a1.

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And now I can solve for a1.

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And what I get is I
get 2m1g minus m2g.

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And I want to divide
through by this denominator.

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And what I have downstairs
is 2m1 plus m2 divided by 2.

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And that's my expression for a1.

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Once I have that
expression for a1,

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I can easily come back
and find out what a2 is.

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Or I can substitute it into
the equation here for a1

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and find out what
the tension is.

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And so I can now easily find
my expressions for a2 and t.

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But I'll leave that
as an exercise.