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# Re: [Phys-L] Rotational Motion Pulley Question

• From: John Denker <jsd@av8n.com>
• Date: Fri, 5 Mar 2021 09:57:22 -0700

On 3/5/21 8:04 AM, Michael Barr via Phys-l wrote:

In Unit 7 of the AP Physics 1 student workbook. Question 7M : Massive Pulley
In the question there is a stationary pulley with a constant mass hanging
off. As the pulley unwinds the radius also decreases.
In part C of the question the answer key explains that rotational
acceleration will decrease as the pulley unwinds due to less Torque. I'm
confused here. Yes there is less torque but if Torque = I x α, and I =
1/2MR^2, then moment of inertiaA decreases even more, so doesn't angular
acceleration go up. A 1/2 drop in radius causes Torque to drop in half but
inertia to drop 4 times. So, doesn't alpha double (assuming we neglect mass
change).

That's an interesting question, especially if we consider what's
behind it.

*) As others have explained, the simple answer is that the stem of
the question specifies that the rope is *light* but thick. To
reinforce that point, it also states:
m(axle) ≫ m(mass) ≫ m(rope)
which would be better stated as:
m(axle) ≫ m(plummet) ≫ m(rope)

*) At some level that's all that need be said, but it is always
good pedagogy to consider where the questions are coming from,
where the misconceptions are coming from.

Here is a plausible hypothesis to consider: Consider the formula
quoted above:
I = ½ M R² [1]
It gives the moment of inertia in terms of "the" radius. Plugging
into this formula leads to trouble, pretty much along the lines
quoted above.

[1] is *not* the definition of moment of inertia. We are much
better off conceptually if we think in terms of:
I = ∫ r² dm [2]

For a *disk* with *uniform* density that reduces to equation [1]
when we integrate from r=0 to r=R ... but in the AP case the overall
radius that enters into the torque calculation is different from
the radius that serves as the limit of integration. We should
integrate only out to the radius of the axle, because there is
negligible m (or dm) beyond there. There is radius beyond there,
but not mass.

So this serves as a nice example of the perils of equation-hunting.
We have equation [1] in terms of R ... but we need to be careful

*) I find the question annoying, because it is physically impossible
to layer a rope or string on top of itself the way the picture
suggests. You would end up with the string winding parallel to
itself, forming a helix along the axle, defeating the point of
the question. You would get no change in radius as the string
unwinds.

They could easily have fixed this bug by describing it as a belt
or strap.

*) Here's an even better way they could have fixed it: There is
such a thing as a /fusee/ which is a conical pulley. For hundreds
of years such things were used to improve the performance of
spring-driven clocks.
https://en.wikipedia.org/wiki/Fusee_(horology)

*) The wording of the question is annoying in additional ways:
-- The rope is sometimes referred to as the string.
-- The title of the question is 7.M Massive Pulley but the
question itself does not use the word pulley. The rope is
wound directly on the axle. This is not how pulleys work.
-- As touched on above, they refer to the falling object as
quote "the mass", but it is not the only thing with the property
of mass, as the term is used in physics. If I were doing it,
I would call this the /plummet/.

It wouldn't kill them to be consistent with their terminology.