Let's look at a
typical application
of Newton's second law
for a system of objects.
So what I want to consider is
a system of pulleys and masses.
So I'll have a fixed
surface here, a ceiling.
And from the ceiling,
we'll hang a pulley, which
I'm going to call pulley
A. And this pulley will
have a rope attached to
it, wrapped around it.
And here we have object 1.
And the rope goes
around the pulley.
And now it's going to go
around another pulley B
and fixed to the ceiling.
So that's fixed.
And hanging from pulley
B is another mass 2.
And our goal in applying
Newton's second law
is to find the accelerations
of objects 1 and 2.
Now, how do we approach this?
Well, the first
thing we have to do
is decide if we're going to
apply Newton's second law, what
is the system that
we'll apply it to?
And there's many different
ways to choose a system.
When we look at
this problem, we'll
have several different systems.
So let's consider the ones
that we're going to look at.
And the first one
is very simple.
It will be block 1.
And the second system
that we look at,
we'll call that AB is pulley A.
Now that brings us to
an interesting question
about pulley B and block 2,
because we could separately
look at pulley B, and we could
separately consider block 2,
or we consider them together.
And I want to first
consider separately pulley B
and block 2.
Now in some ways when you're
looking at a compound system,
and it has four
objects, it makes sense
to apply Newton's second law
to each object separately.
But we'll pay careful
attention to the fact
that object 2 is
connected to pulley B.
And eventually,
we'll see that we
can combine these two things.
So the next step is
once we have identified
our object is to draw a free
body force diagrams for each
of the objects.
So in order to do that,
let's start with object 1.
And we want to consider
the forces on object 1.
Now that brings us
to our first issue
about what types of assumptions
we're making in our system.
For instance, we
have a rope that's
wrapped around this pulley.
And we have two pulleys that
in principle could be rotating.
But what we'd like to do to
simplify our analysis-- so
let's keep track of
some assumptions here.
Our first assumption will be
that the mass, MP, of pulley A
and the mass of pulley B
are approximately zero.
Now the reason for
that is that we're not
going to consider
any of the fact
that these objects have to be
put into rotational motion.
Later on in the course, we'll
see that this will give us
a more complicated analysis.
We're also going to assume
that our rope is not slipping.
So the rope is actually is
just slipping on the pulleys.
So what that means is
it's just the rope is
sliding as the objects move.
Now again, what this
is going to imply
is that the tension in
the rope-- this rope
is also slipping.
And the rope is
massless as well.
It's very light rope.
And all of these
assumptions we've
seen when we analyze
ropes tell us
that the tension T is
uniform in the rope.
So that's our first assumption.
And we need to think
about this before we even
begin to think about the
forces on the object.
And now we can draw our forces.
What do we have?
We have the gravitational
force on object 1.
And now we can identify the
tension pulling in the string,
pulling object 1 up.
Now for every time we
introduce a free body diagram,
recall that we
have to choose what
we mean by positive directions.
And in this case, I'm going to
pick a unit vector down, j hat
1 down.
So that's my positive
direction for force.
Now before I write down
all of Newton's laws,
I'll just write down our
various force diagrams.
So for pulley A,
I have two strings
that are pulling it downwards.
So I have tension and tension.
And this string, I'm
going to call that T2,
is holding that pulley up.
So we have the force diagram.
Now I could write
MAG, but we've assumed
that the pulley is massless.
And again, I'll
call j hat A down.
For object 2, let's
do pulley B first.
Now what are the
forces on pulley B?
I have strings on both sides,
T. Pulley B is massless,
so I'm not putting
gravitational force.
And this string is pulling
B downwards, so that's T3.
And again, we'll write
j hat B downwards.
And finally, I have block 2.
So I'll draw that over here.
I'll write block 2.
In fact, let's say
a little space here.
We'll have j hat B downwards.
Now block 2, what
do we have there?
We have the string
pulling up block 2,
which we've identified as T3.
And we have the gravitational
force on block 2
downward, M2 g.
And there we have j had 2.
So I've now drawn the free body
diagram of the various objects.
And that enables me to
apply Newton's second law
for each of these objects.
So let's begin.
We'll start with object 1.
We have-- remember in all
cases, we're going to apply F
equals m a.
So for object 1, we have
m1g positive downward minus
T is equal to m1 a1.
And that's our F
equals m a on object 1.
So sometimes we'll
distinguish that the
forces we're getting from
our free body diagram.
And A is a mathematical
description of the motion.
For block 2, we have m2g
minus T3 is equal to m2 a2.
And now for pulley A, we
have 2T pointing downwards
minus T2 going upwards.
And because pulley
A is massless,
this is zero even
though pulley A may
be-- it's actually fixed too.
So it's not even accelerating.
And what we see here
is this equation--
I'm going to quickly
note that it tells us
that the string holding pulley
2 up, T2, is equal to 2T.
So we can think of if,
we want to know what
T2 is, we need to calculate T.
And finally we have B. And
what is the forces on B?
We have T3 minus 2T.
And again pulley B is 0.
And so we see that
T3 is equal to 2T.
Now if you think about what
I said before about combining
systems, if we combine
pulley B in block 2,
visually what we're doing
is we're just adding these
to free body diagram together.
When we have a
system B and block 2.
Let's call this j hat downwards.
And when we add these free
body diagram together,
you see that the T3 is now
internal force to the system.
It cancels in pair by
Newton's second law.
And all we have is the
two strings going up,
so we have T and T. And we
have the gravitational force
downward.
And separately, when we
saw that T3 equals 2T
and we apply it there, then
if we consider a system B2,
and look at our free body
diagram, we have m2g minus 2T--
and notice we have the same
result their 2T equals m2 a2.
So in principle now-- and
I'll outline our equations.
We have equation 1.
We have equation 2.
And in these two equations,
we have three unknowns, T, A1,
and A2, but only two equations.
And so you might
think, what about
this missing third
equation here?
However, in this equation,
we have a fourth unknown, T3.
And this equation is just
relating to T and T3.
So in principle, we
would have four unknowns
and three equations.
Or if we restrict our attention
to these two equations,
we have three unknowns
and two equations.
Are unknowns T, A1 and A2.
These are our unknowns.
And now our next step
is to try to figure out
what is the missing
condition that's relating
the sum of these unknowns.
And that will be a
constraint condition
that we'll analyze next.
