Rotation Problem

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lord_indigo
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Rotation Problem

Postby lord_indigo » Wed Dec 08, 2010 11:56 pm UTC

The following is a physics problem from my homework. I've looked back through my notes and through the chapter but I can't seem to find an example or anything of particular use. I searched Google for similar problems and got nothing. If anyone could just sort of point me in the right direction (reminding me what equations I ought to be using would be most helpful) that would be very much appreciated.

The sun's radius is 6.96x10^8m, and it takes 25.3 days to make one rotation. How long will it take to rotate if the sun collapses with no loss of mass to become a neutron star of radius 5 km?

At first I thought it would be a moment of inertia problem but since mass doesn't change.... and we're talking about a change in the radius, so ... maybe...[math]v=r\omega[/math]?

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Re: Rotation Problem

Postby ajd007 » Thu Dec 09, 2010 12:43 am UTC

You're close. You mentioned something about the moment of inertia. What does this depend on?
Also think about the concept of angular momentum. What property does it have in an isolated system?

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Re: Rotation Problem

Postby big boss » Thu Dec 09, 2010 12:43 am UTC

Figure out the initial angular momentum (think moment of inertia of a sphere which depends on radius) then use conservation of angular momenta
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Re: Rotation Problem

Postby Qwert » Thu Dec 09, 2010 12:44 am UTC

ajd007 wrote:You're close. You mentioned something about the moment of inertia. What does this depend on?
Also think about the concept of angular momentum. What property does it have in an isolated system?



In particular, is there something special about the shape of the sun that matters here?
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lord_indigo
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Re: Rotation Problem

Postby lord_indigo » Thu Dec 09, 2010 12:46 am UTC

The sun is treated as a solid sphere, right?

I think it's a conservation of angular momentum problem, so the equation I need is [math]L=I\omega[/math]

Am i getting warmer?


Of course, not forgetting that omega is in radians/sec...If you change this to revolutions per day, then the time would be the reciprocal of this (days/revolution).

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Re: Rotation Problem

Postby Qwert » Thu Dec 09, 2010 1:09 am UTC

What changes in that function when the shape changes?
2 is not equal to 3, not even for large values of 2.
- Grabel's Law
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lord_indigo
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Re: Rotation Problem

Postby lord_indigo » Thu Dec 09, 2010 1:50 am UTC

Qwert wrote:What changes in that function when the shape changes?


Well obviously moment of inertia would change, depending on the shape. (Solid/hollow sphere, solid/hollow cylinder) ... but the shape isn't changing here.
I got a really huge number for my answer (IE, if the sun suddenly collapsed from that Radius to a radius of 5km with no loss of mass it'd make ... like... 765,875,099 revolutions per day.)

.........Is that a reasonable answer?

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Re: Rotation Problem

Postby jjfortherear » Thu Dec 09, 2010 2:41 am UTC

lord_indigo wrote:
Qwert wrote:What changes in that function when the shape changes?


Well obviously moment of inertia would change, depending on the shape. (Solid/hollow sphere, solid/hollow cylinder) ... but the shape isn't changing here.
I got a really huge number for my answer (IE, if the sun suddenly collapsed from that Radius to a radius of 5km with no loss of mass it'd make ... like... 765,875,099 revolutions per day.)

.........Is that a reasonable answer?



The moment of inertia is dependent upon more than the shape - there should be a table in your book with moments of inertia for different shapes (you'll notice they are all functions of something). Your answer looks reasonable, but I'm guessing it'll be a little smaller when you use the right angular momentum eq.
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Qwert
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Re: Rotation Problem

Postby Qwert » Thu Dec 09, 2010 3:03 am UTC

It's what I got plugging into Wolfram Alpha so I think so.

On whether it seems reasonable:

[math]I_{0}\omega_{0} = I_{1}\omega_{1}[/math]

[math]I_{0} = {2\over5}mr_{0}^2[/math]
[math]I_{1} = {2\over5}mr_{1}^2[/math]

[math]{r_{0}^2 \omega_{0} \over r_{1}^2} = \omega_{1}[/math]

with some very rough magnitude checks you get:

[math]r_{0}^2\approx 10^{16}[/math]

[math]\omega_{0} \approx 10^{-6}[/math]

[math]r_{1}^2\approx 10^6[/math]

[math]10^{16-6-6} = 10^4[/math]

[math]{10^4 rad\, sec^{-1}} \approx {10^8 rev\, day^{-1}}[/math]

And [imath]10^8[/imath] is reasonably close to [imath]7.7 * 10^8[/imath] considering the amount of information I completely threw out for that check.
Last edited by Qwert on Thu Dec 09, 2010 3:13 am UTC, edited 3 times in total.
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Re: Rotation Problem

Postby doogly » Thu Dec 09, 2010 3:05 am UTC

And millisecond pulsars are totally things that exist and are awesome.
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Re: Rotation Problem

Postby jjfortherear » Thu Dec 09, 2010 3:07 am UTC

no nanosecond ones?
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Re: Rotation Problem

Postby doogly » Thu Dec 09, 2010 3:12 am UTC

so far, no. probably also impossible since gravitational radiation would take energy away faster than accretion could supply it, though i suppose it might be conceivable.
http://en.wikipedia.org/wiki/Millisecond_pulsar
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Re: Rotation Problem

Postby Tass » Thu Dec 09, 2010 9:52 am UTC

doogly wrote:so far, no. probably also impossible since gravitational radiation would take energy away faster than accretion could supply it, though i suppose it might be conceivable.
http://en.wikipedia.org/wiki/Millisecond_pulsar


Millisecond? Sure
Microsecond? Possibly.
Nanosecond?? Hardly.

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Re: Rotation Problem

Postby Qaanol » Thu Dec 09, 2010 12:38 pm UTC

lord_indigo wrote:The following is a physics problem from my homework. I've looked back through my notes and through the chapter but I can't seem to find an example or anything of particular use. I searched Google for similar problems and got nothing. If anyone could just sort of point me in the right direction (reminding me what equations I ought to be using would be most helpful) that would be very much appreciated.

The sun's radius is 6.96x10^8m, and it takes 25.3 days to make one rotation. How long will it take to rotate if the sun collapses with no loss of mass to become a neutron star of radius 5 km?

At first I thought it would be a moment of inertia problem but since mass doesn't change.... and we're talking about a change in the radius, so ... maybe...[math]v=r\omega[/math]?

Are you expected to assume that the sun’s density is both spherically symmetric, and the mass distribution of the neutron star will be similar (up to scaling) to that of the main sequence star? Or, for that matter, that the whole sun rotates as a unit? Seems kind of a stretch to me.
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Re: Rotation Problem

Postby LucasBrown » Thu Dec 09, 2010 2:12 pm UTC

Problems like the one above are frequent occurrences in high school physics textbooks (there are no less than five of them, in various guises, in mine), and expecting a high school physics student to consider the mass distributions inside the Sun and a neutron star is really too much of a stretch unless they're given beforehand. My physics teacher would never give a problem in which the mass distribution was nonuniform unless it was a very simple density function on a thin rod.

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Re: Rotation Problem

Postby doogly » Thu Dec 09, 2010 2:21 pm UTC

It's just problematic that the approximation is so resoundingly shitty. Assume the earth has uniform density, sure, not an awful first order thing to do. Assume the sun has uniform density and that it doesn't expel any mass in a transition to neutron star, what the fuck. A reasonable approximation should get you the right order of magnitude. Off by a few oom's, well, maybe in physics 1 you can put up with that. Getting the end state of stellar evolution wrong? No, don't do that.
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Re: Rotation Problem

Postby gmalivuk » Thu Dec 09, 2010 5:56 pm UTC

The assumptions made are that it doesn't expel any mass (which, considering we're talking about the already impossible scenario of our little Sun collapsing into a neutron star, doesn't seem too terrible), and that its mass distribution is maintained (proportionally) as it shrinks. Sure, not an entirely reasonable assumption, but not nearly as crazy as assuming completely uniform distribution.

So if the radius decreases by a factor of about 140,000, it means the rotational velocity will increase by the square of that, or about 20 billion.
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Re: Rotation Problem

Postby doogly » Thu Dec 09, 2010 6:58 pm UTC

Yeah, it's highly problematic. Neutron stars have a totally different EOS (not that we know what it is, but still, it's not even the same as white dwarf) and one spinning this fast isn't even going to be such a great sphere. I call shenanigans.

But neutron stars are pretty cool stuff, so putting them into the otherwise boring ramp and pulley shit you get in intro physics does keep business lively. So I won't complain altogether, just keep the bit complaining I just did.
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Re: Rotation Problem

Postby KittenKaboodle » Sat Dec 18, 2010 6:06 am UTC

Qaanol wrote:Are you expected to assume that the sun’s density is both spherically symmetric, and the mass distribution of the neutron star will be similar (up to scaling) to that of the main sequence star? Or, for that matter, that the whole sun rotates as a unit? Seems kind of a stretch to me.


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