GREPhysics.NET: GR8677 #2

GR8677 #2

Problem

GREPhysics.NET Official Solution

Mechanics $\Rightarrow$ }Vector

While there may be a more quantitative solution, the simplest solution is qualitative, based on elementary vector addition and knowledge of the force center.

The problem states that the object orbits the Earth in a perfect circle, initially. This means that the initial velocity ( $\vec{v_0}$ ) is perpendicular to the vector pointing to the earth center ( $\vec{n}$ ), i.e., it's tangent to the circular path. This is the condition for uniform circular motion (the centripetal acceleration is due to the Gravitational Law).

After firing a missile straight to Earth center, its velocity gains an extra normal component ( $-\vec{v_n}$ ), equal and opposite to the velocity of the missile fired to Earth. Thus, its trajectory would deviate from the circular trajectory.

Because the only source of acceleration comes from the Earth center, $-\vec{v_n}$ , which is parallel to the centripetal acceleration provided by the Earth, will eventually go to 0. Recall that acceleration does not effect velocity components in the perpendicular direction (to wit: a projectile fired on Earth has the same constant $v_x$ , but its $v_y$ changes). There will thus always be a (nearly constant) tangential velocity, even at the perigees. However, $-\vec{v_n}$ will go to 0 at the perigees. The tangential velocity will remain more-or-less constant, so that instead of spiraling inwards, the path becomes an ellipse, as $-\vec{v_n}$ is restored at the apogees and zero'ed at the perigees.

(In a more down-to-earth form, this problem is essentially a projectile firing question with no numerical work involved.)

Alternate Solutions

angiep
2005-11-09 07:48:42

I disagree with your answer. It's a question about the classification of orbits. You need a graph of the effective potential. I found some stuff on it here: http://www.bun.kyoto-u.ac.jp/~suchii/eff.potent.html. I like how it's treated better in Goldstein (classical mechanics, i have the 3rd ed). Basically, the only orbits possible with an inverse square force law are a hyperbola, parabola, ellipse, or circle. Circular only occurs at a particular radius. A perturbation from there gives you a slightly higher energy, and you are in a region of an elliptical orbit.

I like your site! Thanks!

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Comments

Patrusfarr
2009-09-28 18:57:20

Someone explain to me why it cannot possibly be a hyperbola. Is it because we said the words "slightly perturbed?" Given enough boost, an object will go into a hyperbolic orbit.

kroner
2009-09-28 19:35:14

Yeah that's right. To go from a circular orbit to a hyperbolic orbit you need to more than double the kinetic energy of the satellite, which is a big change in the context of any realistic Earth orbit.

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engageengage
2009-01-17 17:58:09

The reason that it cannot be E is because a perturbation would cause oscillations to occur at the same frequency as the initial orbit. Therefore, it would not cause 'many' radial oscillations, but only one, per revolution. In fact, if you think about the inner and outer radii of a perturbed orbit where the radial oscillations match the orbital 'oscillations', you see that the orbit is actually ellipse.

It might be possible to do the derivation in a couple of minutes, if one is to know exactly what to look for. Consider the effective potential:

$\gamma = GM\mu$

$U = \frac{-\gamma}{r}+\frac{l^{2}}{2\mu r^{2}}$

$\frac{dU}{dr} = \frac{\gamma}{r^{2}}-\frac{l^{2}}{\mu r^{3}}$

the stable orbit occurs when this quantity is zero, i.e., this is an equilibrium point.

$r_{eq}=\frac{l^{2}}{\gamma \mu}$

We take the second derivative, knowing that we can expand the potential using a taylor series, approximating the potential as a quadratic potential well. Here, the second derivative of the potential evaluated at the equilibrium point corresponds to the force constant, 'k' in a typical harmonic oscillator problem (write out the taylor series to see why).

$\frac{d^{2}U}{dr^{2}} = \frac{-2 \gamma}{r^{3}}+3\frac{l^{2}}{\mu r^{4}}$

At $r_{eq}$ :

$k = \frac{d^{2}U}{dr^{2}} = \frac{\gamma}{r_{eq}^{3}}$

Now, we can easily deduce the the frequency of the oscillation, plugging in the value of gamma:

$\omega^{2}=\frac{k}{\mu}=\frac{GM}{r_{eq}^{3}}$

Finally, where T is the period:

$T^{2} = (\frac{2\pi}{w})^{2} = \frac{4\pi^{2}r_{eq}^{3}}{GM}$

This is just Kepler's Third Law for orbits! Therefore, any radial perturbation would cause the radial oscillations equal to the angular period. So, the answer cannot be E, and must be A

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dcn
2008-03-26 07:59:22

You have a typo in your explanation. I'm not the greatest speller but I think in the fourth paragraph, second sentence, it should be "affect" instead of "effect". Effect is a noun and is the result when something is affected, which is a verb. This is a great site. Thanks.

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climbing_evergreen@grephysics.net
2007-11-23 01:28:29

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Richard
2007-11-02 09:30:28

Right, I understand all that, but what is Goldstein talking about in Chapter 3.6?
Look at Figure 3.13. And there is this:
"If a circular orbit is stable, then a small increase in the particle energy above the value for a circular orbit results in only a slight variation of $r$ about $r_0$ . It can be easily shown that for such small deviations from the circularity conditions, the particle execute a simple harmonic motion..."
So I would think to choose (E).

But I've only given Goldstein a cursory read, so I'm probably missing something. It seems that if you look at the potential curve, it's pretty obvious when you get a circular orbit (at the base where only one $r$ is possible) and when you get an elliptical orbit (anywhere where you have a finite range of possible $r$ 's, which occurs at total energy less than zero.)
So how do I reconcile these two thoughts?

physicsisgod
2008-11-05 16:44:09

I think the solution to this confusion is that the oscillations in the orbit occur when we slightly perturb the radius, but we leave the kinetic energy constant. So really, we're increasing the energy of the particle by changing its potential energy (we move it in the gravitational field, but keep its momentum constant). In the case of this GRE problem, we are acutally adding to the particle's momentum, by giving it some radial velocity for a short period of time. I think this might be a better way to think about the difference.

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phillipkp
2006-04-30 09:42:27

Check out section 8.7 in Marion and Thornton also.

kolahalb
2007-12-07 11:44:02

I would suggest something...

After a small perturbation,energy is more and we get no more circular orbit.For inverse square law forces,(and for F=-kr type of forces) we get stable noncircular orbits(which are closed).So,correct answer is (A)

Now why not (E)?I strongly suggest to refer to Atam P Arya's book.To have radial oscillation (and an ultimately closed trajectory) we must have
T(r)/T(theta) =a rational number.[The figure is a popular one---front cover page of Goldstein].
when the ratio=1,we get T(r)=T(theta) and ellipse results.That is what exactly happens in attractive inverse square alw force.

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angiep
2005-11-09 07:48:42

yosun
2005-11-09 14:24:05

As angiep suggests, this problem can also be solved via classification of orbits. More on this method can be found in the solutions for other problems on this site:

GR9677 66; http://grephysics.net/disp.php?yload=prob&serial=2&prob=66 />
GR9677 23; http://grephysics.net/disp.php?yload=prob&serial=2&prob=23 />
GR0177 22;

http://grephysics.net/disp.php?yload=prob&serial=3&prob=22 />

sharpstones
2007-04-01 13:54:27

Just to add some more detail since this is an important concept to have down.

Classification of Orbits by total energy $E_t$
$E_t > 0 $Hyperbolic orbit<br />$ E_t = 0 $Parabolic orbit<br />$ E_t < 0$ -> Elliptical Orbit
-> special case $E_t = \frac{-GMm}{2R}$

So changing your energy from a circular orbit will give you an ellipse... increase energy too much and you can eventually reach a hyperbolic orbit.

sharpstones
2007-04-01 13:56:12

Just to add some more detail since this is an important concept to have down.

Classification of Orbits by total energy $E_t$
E_t > 0 Hyperbolic orbit
E_t = 0 Parabolic orbit
E_t < 0 Elliptical Orbit
-> special case $E_t = \frac{-GMm}{2R}$

So changing your energy from a circular orbit will give you an ellipse... increase energy too much and you can eventually reach a hyperbolic orbit.

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