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*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.

This leads to an interesting point about the physics: Equation

[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

about /which/ R.

*) 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.

**Follow-Ups**:**Re: [Phys-L] Rotational Motion Pulley Question***From:*Brian Whatcott <betwys1@sbcglobal.net>

**References**:**[Phys-L] Rotational Motion Pulley Question***From:*Michael Barr <zubarsky@gmail.com>

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