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How do we solve problems
involving massive pulleys using

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Newton's laws?

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As a simple example, let's
look at this problem,

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consisting of a block
of mass m1 hanging

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from a massive pulley that
has a moment of inertia I

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and radius r.

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We'll find the acceleration
of the block, a1,

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and the angular acceleration
of the massive pulley, alpha.

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It is critical to remember
that the first step in solving

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these problems is to define
the positive x and y directions

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and the positive
direction of rotation.

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In two-dimensional
problems, you're

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free to find any
direction to be positive.

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Here, we'll set our positive
x and positive y like this.

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Once we pick x and
y, the direction

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of the positive rotation
is now also defined

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by the right-hand rule.

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You'll see shortly that
defining positive directions

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at the beginning will save you
from a negative sign nightmare

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later on.

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Now let's break down Newton's
laws for the different parts

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of the system.

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First, for the block, we will
write down the linear version

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of Newton's second law.

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For the sum of forces, we
have gravity pointing down

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and tension pointing up.

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According to the convention we
just defined, T1 is positive

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and m1g is negative.

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Notice that since the
block is accelerating,

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T1 minus m1g is not 0
but is equal to m1a1.

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Notice that here we've set the
signs for tension and weight,

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but we don't yet know which
were the block is accelerating.

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So we just start
by writing m1a1.

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And if, in the end, we
get that a1 is negative,

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we'll know that it's
actually accelerating

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in the negative direction.

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Now let's look at the pulley.

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Newton's second law
in its rotational form

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is tau is equal to I alpha.

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For a normal pulley,
the string is always

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tangential to the
side of the pulley.

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So r and f are
perpendicular, so that means

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the torque is just T1 times r.

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What's the sign of this term?

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Well, according to the positive
direction that we defined,

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this torque is positive.

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Again, we'll leave the
sign of the I alpha term

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to be positive.

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And alpha will turn out to be
either positive or negative.

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Finally, we need to connect
these two equations.

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Because the block is
connected to the pulley using

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an ideal, taut,
inextensible rope,

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a1 is going to be
related to alpha.

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We just need to figure out
exactly how they're related.

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In other words, we need to write
down the constraint condition.

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Let's say that the
block hypothetically

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is moving upwards.

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In this case, what's
happening to the pulley?

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It must be spinning
clockwise to pull the rope up

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as the block goes in, so it has
a clockwise angular velocity,

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which, according to our
convention, is negative.

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In fact, if the pulley
rotates a full turn

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in the clockwise
negative direction,

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the angle changes
by negative 2 pi,

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and 2 pi times r of the
rope will be pulled up.

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So we can write that delta y is
equal to negative r delta theta

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or, taking some derivatives,
a1 is equal to negative r

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times alpha.

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After setting the
sign conventions,

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writing Newton's laws for
different parts of the system,

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and then writing down
the constraint condition,

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we have three equations
and three unknowns

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for which we can now solve.