Rotation of a sphere because of its gravitational field

[ShayanJ]

I want to know how can I prove that a massive sphere rotates becuase of its gravitational field.I thought about it and did the following:
The gravitational force applied to a mass m which is on the axis prependicular to the plane of a hollow ring and passing through its center is as follows
${\it dF}={\frac {G{\it dM}\,mz}{ \left( {r}^{2}+{z}^{2} \right) ^{3/2}}}$
If I integrate the equation above from z=0 to z=2R I should get the force applied to a point on a hollow sphere and I get:
$F=1/8\,{\frac {GMm\theta\,\ln \left( z \right) \sqrt {2}}{\pi \,{R}^{2}}}$
$\theta$ is from 0 to $2 \pi$ and z from 0 to 2R.But as you can see,at z=0,it becomes infinity.So I wrote the taylor series of ln(z).But my calculations for $\omega$ didn't yeild 7.3 * 10^(-5) which is the angular velocity of earth.What's wrong?
thanks

 

[Doc Al]

I want to know how can I prove that a massive sphere rotates becuase of its gravitational field.

Why would you think that it would rotate? Explain what you mean.

 

[ShayanJ]

Earth rotates around itself and if you think,the only reason can be its own gravitational field.

 

[Tea Jay]

Earth rotates around itself and if you think,the only reason can be its own gravitational field.

What about the fields of other bodies that can influence it, such as the moon or sun?

What about the motion of its crust or mantle?

 

[lmcelroy]

There are many factors that influence the Earth's rotation, none of which is its mass. Check this out: http://en.wikipedia.org/wiki/Earth's_rotation#Origin

 

[D H]

Earth rotates around itself and if you think,the only reason can be its own gravitational field.

No! The Earth rotates now because it was rotating in the past and because angular momentum is a conserved quantity.

Newton's first law, a body persists in its state of motion unless acted upon by an external force, is in a sense a statement regarding conservation of linear momentum. There's a direct analog of Newton's first law for rotational motion, a body persists in its state of rotation unless acted upon by an external torque.External torques from the Moon, the Sun, and the other planets do act on the Earth. However, these external torques are small. Because they are small, they can be treated as being effectively zero for short periods of time (years and even decades are short periods of time in this context). Earth's angular momentum is more or less constant over the span of years or decades. It's only on the scale of centuries or longer (or on the scale of extremely precise measurements) that the Earth's angular momentum can not be viewed as being constant.

 

[ShayanJ]

Thanks for the answers.
I now have a psychological theory about undergraduate students(like myself).
This question came into my mind when our professor was teaching the central forces.
Looks like we just get excited when we learn sth new and stick everything to that new concept.

 

[jetwaterluffy]

Earth rotates around itself and if you think,the only reason can be its own gravitational field.

No the only reason is the conservation of angular momentum. Although a similar effect does possibly exist: http://en.wikipedia.org/wiki/Gravitomagnetism but the effect is tiny.

 

[Buckleymanor]

No! The Earth rotates now because it was rotating in the past and because angular momentum is a conserved quantity.

So when the dust etc, was pulled together when it was forming by the gravity of the main body, angular momentum caused it to rotate.
Won't rain falling do the same.

 

[D H]

So when the dust etc, was pulled together when it was forming by the gravity of the main body, angular momentum caused it to rotate.

Correct. Whatever angular momentum that that infalling stuff already had, plus whatever angular momentum that the proto-Earth already had, was conserved.

Won't rain falling do the same.

Not really. First off, rain doesn't have much angular momentum. Much more importantly, whatever angular momentum rain does add to the Earth comes from the Earth+oceans+atmosphere system, not from outside. Rain does not add angular momentum to the Earth+oceans+atmosphere system.

There is an ongoing but small transfer of angular momentum from the Earth to the Moon. This is what causes the Earth's rotation rate to slowly decrease and the size of the Moon's orbit to slowly increase. However, this transfer is very slow. Over short spans of time (seasons, years, even decades), the angular momentum of the Earth+oceans+atmosphere is more or less constant. There is seasonal transfer of angular momentum between the Earth+oceans and the atmosphere, but this is cyclical and bounded. There is no secular change.

 

[jetwaterluffy]

So when the dust etc, was pulled together when it was forming by the gravity of the main body, angular momentum caused it to rotate.
Won't rain falling do the same.

It starts as a large cloud of dust spinning very slowly. As the size of it decreases, the spinning speed speeds up to compensate. This can't happen with rain, as rain doesn't decrease the size of the earth.

 

[D H]

It starts as a large cloud of dust spinning very slowly. As the size of it decreases, the spinning speed speeds up to compensate.

That's not a good model of how stars form, let alone planets. The Sun has 99.9% of the mass of the solar system, but only 1% of the angular momentum. This is the angular momentum problem.

Assuming the total angular momentum of the cloud with respect to its center of mass is non-zero, the formation of the protostar will act to flatten the cloud into a disc. With the formation of the disc, the material that falls into the protostar is that material that has little angular momentum. The formation of the disc solves the angular momentum problem.

 

[jetwaterluffy]

That's not a good model of how stars form, let alone planets. The Sun has 99.9% of the mass of the solar system, but only 1% of the angular momentum. This is the angular momentum problem.

Assuming the total angular momentum of the cloud with respect to its center of mass is non-zero, the formation of the protostar will acts to flatten the cloud into a disc. With the formation of the disc, the material that falls into the protostar is that material that has little angular momentum. The formation of the disc solves the angular momentum problem.

Erm.. sorry can you rephrase that? Maybe I didn't get it when I was told, and subconsciously made something up to compensate, so I'd like you to explain it again for me. (the above post to me doesn't make sense.)

 

[phinds]

Thanks for the answers.
I now have a psychological theory about undergraduate students(like myself).
This question came into my mind when our professor was teaching the central forces.
Looks like we just get excited when we learn sth new and stick everything to that new concept.

That is an excellent observation.

It's another way of saying the old saw "to a man with a hammer, all problems look like nails"

 

[D H]

Erm.. sorry can you rephrase that? Maybe I didn't get it when I was told, and subconsciously made something up to compensate, so I'd like you to explain it again for me. (the above post to me doesn't make sense.)

Starting over, then.

It starts as a large cloud of dust spinning very slowly. As the size of it decreases, the spinning speed speeds up to compensate.

That is not a good model for how planets or stars form. It fails to explain why even though the Sun has 99.9% of the total mass of the solar system, it only has 1% of the solar system's total angular momentum. This is called the angular momentum problem.

A better model is the solar nebula disc model. The star and planets eventually form from some nebular cloud. Assume that the total angular momentum of this cloud with respect to the cloud's center of mass is non-zero. The star starts forming due to some uneven mass distribution within the cloud. As the protostar accumulates more mass, the increasing gravitational force will cause the nebular cloud to flatten out and form a rotating disc. The dust that has angular momentum with respect to this protostar will tend to end up somewhere on the disc. The dust that has near-zero angular momentum is what falls into the growing star. This nicely solves the angular momentum problem.

This spinning cloud does not explain planet formation, either. Planets form after the disc has formed. The orbital velocity of an orbiting object is $\sqrt{G(M+m)/r}$. A growing protoplanet will have a slightly greater orbital velocity than will the dust with which it is co-orbiting. The growing protoplanet will plow through this dust cloud, gathering mass and clearing the vicinity as it does. The growing protoplanet also migrates inward, encountering new dust. The density of the material is thus slightly greater in the direction of the protostar. From the perspective of the planet, it is encountering a wind of dust that is slightly denser toward the star.

This is a fairly good explanation of the formation of gas giants, but not quite so good for the rocky planets. There are still some unknowns left in explaining the formation of the rocky planets. The Earth, with its huge Moon, is particularly problematic. The current hypothesis is that a Mars-sized protoplanet collided with the proto Earth shortly after the formation of the solar system was more or less complete.

 

[Buckleymanor]

Not really. First off, rain doesn't have much angular momentum. Much more importantly, whatever angular momentum rain does add to the Earth comes from the Earth+oceans+atmosphere system, not from outside. Rain does not add angular momentum to the Earth+oceans+atmosphere system.

I imagined that the energy was provided from outside the Earth+oceans+atmosphere system by the Suns evaporation of water.

There are still some unknowns left in explaining the formation of the rocky planets. The Earth, with its huge Moon, is particularly problematic. The current hypothesis is that a Mars-sized protoplanet collided with the proto Earth shortly after the formation of the solar system was more or less complete.

With regards to the explanation of the formation the Moon.If there was a large collision you would expect the Moon to be spinning because of it and at more of a rate than the Earth.
How come it has not been able to maintain it's spin from the impact you would expect it to be conserved.

 

[Drakkith]

I imagined that the energy was provided from outside the Earth+oceans+atmosphere system by the Suns evaporation of water.

With regards to the explanation of the formation the Moon.If there was a large collision you would expect the Moon to be spinning because of it and at more of a rate than the Earth.
How come it has not been able to maintain it's spin from the impact you would expect it to be conserved.

The moon is not spinning as fast as it once was because of tidal locking.
http://en.wikipedia.org/wiki/Tidal_locking

 

[D H]

I imagined that the energy was provided from outside the Earth+oceans+atmosphere system by the Suns evaporation of water.

That makes no sense.

With regards to the explanation of the formation the Moon.If there was a large collision you would expect the Moon to be spinning because of it and at more of a rate than the Earth.
How come it has not been able to maintain it's spin from the impact you would expect it to be conserved.

As Drakkith already noted, there's tidally locking to consider.

 

[Buckleymanor]

The moon is not spinning as fast as it once was because of tidal locking.
http://en.wikipedia.org/wiki/Tidal_locking

Interesting though it seems unclear if the Moon should speed up it's rotation if it's not spinning as fast as it was.

From the cited wikipedia article,

The angular momentum of the whole A-B system is conserved in this process, so that when B slows down and loses rotational angular momentum, its orbital angular momentum is boosted by a similar amount (there are also some smaller effects on A's rotation). This results in a raising of B's orbit about A in tandem with its rotational slowdown. For the other case where B starts off rotating too slowly, tidal locking both speeds up its rotation, and lowers its orbit.

There seems to be a contradiction when B starts of rotating too slowly how does the Moon Know it started rotating at a faster speed than it is now.
For all purposes when it slows down it won't "know" if it started faster or that this was it's starting speed.

 

[D H]

There seems to be a contradiction when B starts of rotating too slowly how does the Moon Know it started rotating at a faster speed than it is now.
For all purposes when it slows down it won't "know" if it started faster or that this was it's starting speed.

There's no contradiction, and the Moon doesn't need to "know" anything. (How can it?) It's all in the math, and that math is only concerned with what is happening now.

 

[Buckleymanor]

There's no contradiction, and the Moon doesn't need to "know" anything. (How can it?) It's all in the math, and that math is only concerned with what is happening now.

Which was my point, if the Moon does not know anything and the math is only concerned with what is happening now it could speed up tomorrow.

 

[D H]

No, it can't. Now that the moon's rotation rate is in sync with the orbital rate, it will stay that way. The orbital rate is decreasing thanks to the ongoing transfer of angular momentum from the Earth's rotation to the orbit of the Earth and moon about one another.

 

[Drakkith]

Which was my point, if the Moon does not know anything and the math is only concerned with what is happening now it could speed up tomorrow.

Only if something causes it to speed up. The interactions with Earth is not, and will not cause this to happen. Furthermore I am...confused with your statement "if the Moon does not know anything". Inanimate objects aren't capable of "knowing" anything. They merely react to various forces. The interactions of these forces are what is causing the changes.

 

[Buckleymanor]

Only if something causes it to speed up. The interactions with Earth is not, and will not cause this to happen. Furthermore I am...confused with your statement "if the Moon does not know anything". Inanimate objects aren't capable of "knowing" anything. They merely react to various forces. The interactions of these forces are what is causing the changes.

Well maybe not some time soon but there must be a possibility that it could speed up in the future.
As energy is transferred from the Earth to the Moon the orbit of the Moon increases[distance].
This will enable the Moon to be less locked as the gravitational pull will decrease on the locked side.
Which could result in speeding.
As for the Moon knowing something unlike other inanimate objects it has a man in it.