# Will it revolve?

#### vinter

There is a lonely massive star standing almost fixed at some point in this universe. There is a small planet projected with some velocity in some direction near the star. Given the masses of the star and the planet and the velocity, can you find whether the planet will collide with the star?
If yes, how?
(Neglect the displacements of the star)
One thing that I thought and later realised that it was wrong was this :-
If the planet has a non-zero angular momentum initially, from the conservation of angular momentum, it will never collide, because if it did, it's angular momentum would become zero at the time of collision which is not possible. But I realised that it's angular momentum can remain non-zero during collision if it's velocity becomes infinite.
So the problem is still not solved.

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#### Janus

Staff Emeritus
Gold Member
vinter said:
One thing that I thought and later realised that it was wrong was this :-
If the planet has a non-zero angular momentum initially, from the conservation of angular momentum, it will never collide, because if it did, it's angular momentum would become zero at the time of collision which is not possible. But I realised that it's angular momentum can remain non-zero during collision if it's velocity becomes infinite.
So the problem is still not solved.
If the planet has a non-zero angular momentum, it can still collide with the star. It will just strike the star off-center. In such a case it will transfer its angular momentum to the star, and the angular momentum of the system is conserved.

#### DaveC426913

Gold Member
vinter said:
If the planet has a non-zero angular momentum initially, from the conservation of angular momentum, it will never collide, because if it did, it's angular momentum would become zero at the time of collision which is not possible. But I realised that it's angular momentum can remain non-zero during collision if it's velocity becomes infinite.
Only if the star and planet are point objects. In reality, their centres of mass can be hundreds of thousands of miles apart and they can still collide.

#### vinter

OK, assume that they are point objects. That was the actual problem. I missed to mention it.

#### mtong

I would consider it a “projectile motion” problem; however one must take into the acceleration that the gravitation field of the star would provide, in both the direction of the star and in the direction of motion attributed to the moving planet.

#### vinter

Consider it any kind of a problem. Just tell me is there any simple way to predict that it will collide or not or do we have to find the exact motion profile and see whether it passes through the star?

#### Janus

Staff Emeritus
Gold Member
If they are point particles, then unless their intitial velocities are pointed directly at each other or they intially are motionless with respect to each other, they will not collide.

#### toocool_sashi

i think that the motion wud be like the 1st "flaw" in rutherfords atomic model if u remember..."electrons moving with acceleration wud radiate energy continously and they wud finally follow a spiral orbit and collide with the nucleus and the atom wud collapse". Maybe the planet revolves like that...in a spiral way...always maintaining a component of velocity NOT in the direction of line joining CoM of both the masses...such that angular momentum is non zero.

#### vinter

toocool_sashi said:
i think that the motion wud be like the 1st "flaw" in rutherfords atomic model if u remember..."electrons moving with acceleration wud radiate energy continously and they wud finally follow a spiral orbit and collide with the nucleus and the atom wud collapse". Maybe the planet revolves like that...in a spiral way...always maintaining a component of velocity NOT in the direction of line joining CoM of both the masses...such that angular momentum is non zero.

The flaw in the Rutherford's model arises because the revolving particle is CHARGED and accelerated charged particles lose energy. Here, the planet is not charged and so what you are saying won't happen.

But I think what Janus says is correct because of the following reason :-

The angular momentum of the planet about the star must be conserved. If the planet collides with the star, at the time of collision it will have a zero angular momentum unless its velocity becomes infinite. But if its velocity is infinite, then its KE is also infinite and that doesn't obey the law of conservation of energy since it had a finite ME initially.

#### toocool_sashi

no no yaar...im not saying that it follows rutherfords FLAW coz its charged...im just saying that the planet might move spirally towards the star...i just quoted rutherfords flaw as an example coz the 1st image that came 2 my mind when i thought of a spiral path of the plant and collision into the star was electron - atom system ;) thats all..im not saying that planet radiates energy continously and loses KE coz its not charged 1st of all.
As far as assuming that they are point masses goes...frankly i have always believed that we shud make such assumptions a bit carefully...suppose i say consider 2 spheres to be point masses(we CAN do that..its proved and u all know it)..place them together(they are touching)..and if i ask u what is the gravitation force of attraction between them...ur answer wud be infinite coz if they are pointmasses and distance between CoM wud be zero...but in reality ...they are not point masses and dist between CoM wud be non zero(however small...but NON ZERO) so the force is not infinity....ok im deviating from the point here
I will now try 2 give the full solution to this prob....given the masses of star and planet and velocity of planet....consider the motion of the planet at any arbit distance , using conservation of energy, we can find the velocity of the planet. If the radial component of this velocity is enough 2 balance the centripetal force(gravitation) at that distance, then the planet will start revolving...or else...it will go closer and closer to the planet...if at no distance can CPF be balanced....it will collide with the star and both will revolve as 1 single body and the angular speed of revolution can be calculated using conservation of ang. momentum......do u notice any flaw in this explanation?

#### vinter

Yes, I do notice some flaws.
First of all, using conservation of energy, we can only find the magnitude of the velocity; we can't say anything about its radial component.
Then, I don't understand what you mean by the radial component being enough to balance the centripetal force. If you mean gravitational pull = mv(squared)/r, then, you are wrong, because of several reasons one of them being the v in the above expression should not be the radial component.

#### toocool_sashi

hmm...yes....ive become dumb. lol...i meant normal component of velocity there sorry. and well...to find the components of velocity...actually the normal component of velocity will not change coz there no force perpendicular to line joining CoM so i dont think normal component of velocity wud change so we just have 2 find out where it will balance the gravitational pull(mv^2/r). But my explanation wudnt work...cos along with the normal component...there is the radial component which is pulling it towards the star and it WILL move towards the star in a spiral fashion no matter what...i think i can make this statement...3 cases (1) u provide velocity ONLY in the normal direction and it is exactly equal 2 complete circular motion with that radius...circular motion will occur
(2) any velocity along the line joining CoM only...or no velocity at all...headon collision will occur and no rotation motion occurs afterwards...purely translational collision
(3) velocity obliquely...it will undergo motion of type of planet-star...i mean eleptical with star @ one of its foci...it may collide with star during its motion due to physical dimensions of star.
any flaws now??

#### vinter

case (3) doesn't seem to be that correct. Recently i wrote a computer simulation for gravitation force. It just used the the gravitation force law to predict the motion of a planet if the initial distance and velocity were given. And I found several different kinds of motion. In most of the cases, the following would happen :-
the planet would start in sort of an elliptical orbit but when it would be about to reach the starting point, you would realise that it wasn't going to retrace the path exactly. It would go along a slightly rotated ellipse and this would keep going on. In fact, in very very few cases did the velocity and the distance match to make the planet revolve.

But why aren't you satisfied with my explanation? The one involving angular momentum and energy? I think it gives a rigorous and complete solution to the problem!

#### toocool_sashi

vinter said:
the planet would start in sort of an elliptical orbit but when it would be about to reach the starting point, you would realise that it wasn't going to retrace the path exactly. It would go along a slightly rotated ellipse and this would keep going on. In fact, in very very few cases did the velocity and the distance match to make the planet revolve.
Yes ive heard that even our solar system's planets dont trace exact orbits, rather, their orbits keep changing ever so slightly every time they finish a revolution. But with our (12th grade) knowledge of gravitation can we predict that?

vinter said:
But why aren't you satisfied with my explanation? The one involving angular momentum and energy? I think it gives a rigorous and complete solution to the problem!
yaar i dont read solutions posted by ppl, i just read the question and directly post my answer. Maybe thats why u mite have seen in other posts, i give the answer after @least 10 ppl who have posted their answer earlier

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