Gravity problem

[ManOfSteel]

Hello,

 

Suppose there are two equally massive objects and a third object in between. Each of the two objects is pulling the third of the middle in an opposite direction. Does the gravitational force of one of the two objects cancel the gravitational force of the other? Is there any loss in mass? Or gain in energy?

 

Why are two objects attracted to each others?

 

Thank you in advance.

 

PS: I'm not sure I posted this in the right forum as it is both related to classical physics and general relativity. So moderators, feel free to move it.

[Klaynos]

This looks a bit homeworky...

 

I'll deal with this strictly using newtonian gravity.

 

To calculate the motion of the object in the middle you need to consider the net force acting on it.

 

Force due to gravity is:

 

[math]F=gm=\frac {-GMm} {r^2} \hat r[/math]

 

For the force due to one of them the r hat will be +ve and the other -ve, and for the net force you just add them all up, so do you think they cancel?

 

As to why there is gravity well considering GR you'd talk about bending space time, considering QM you'd probably talk about gravitons...

[timo]

Suppose there are two equally massive objects and a third object in between. Each of the two objects is pulling the third of the middle in an opposite direction. Does the gravitational force of one of the two objects cancel the gravitational force of the other?

Depends on the exact positions. You probably meant that the 3rd object is exactly in the middle of the two other point-sized (or spherical) objects, in which case the gravitational force on the 3rd object would indeed be zero (forces exerted on an object due to different causes add up). The two "outer" masses would feel a net force due to the other objects, though.

 

Generally, the gravitational force acting on a point-sized object at position [math] \vec x[/math] of mass m caused by a point-sized (or spherical) object with mass M at position [math]\vec X[/math] is [math] \vec F_{mM} = G \frac{mM}{\| \vec x - \vec X \| ^3}\left( \vec X - \vec x \right)[/math], with G being the gravitational constant. You can compute the forces for different scenarios yourself by considering the relevant forces acting on an object.

 

 

Is there any loss in mass? Or gain in energy?

Depends on what exactly you are looking at and how you define either of the two. I'm inclined to say "no", but you could possibly explain why you think there could be to get some better replies.

 

Why are two objects attracted to each others?

There can natually never be a definite answer to such a question. You could always be the follow-up question "why that answer?". We simply do observe that massive objects attract each other, have found a way to describe this attraction quantitatively, and call that mechanism "gravity".

 

PS: I'm not sure I posted this in the right forum as it is both related to classical physics and general relativity. So moderators, feel free to move it.

I think you should restrict yourself to Newtonian gravity; no need for unnecessary complification, I think.

 

EDIT: @Klay: No, I'm not senile; I didn't see you're on

EDIT2: Thx@ydoaps.

[ydoaPs]

Generally, the gravitational force acting on a point-sized object at position [math] \vec x[/math] of mass m caused by a point-sized (or spherical) object with mass M at position [math]\vec X[/math] is [math] \vec F_{mM} = G \frac{mM}{\| \vec x - \vec X \| ^3}\left( \vec X - \vec x \right)[/math], with G being the gravitational constant. You can compute the forces for different scenarios yourself by considering the relevant forces acting on an object.

 

fixed

[ManOfSteel]

Thank you for the fast replies.

 

 

Klaynos,

This looks a bit homeworky...

My post? No no, don't worry! I was motivated to post this because of a discussion on another forum.

 

Force due to gravity is:

 

[math]F=gm=\frac {-GMm} {r^2} \hat r[/math]

 

For the force due to one of them the r hat will be +ve and the other -ve, and for the net force you just add them all up, so do you think they cancel?

Could you give me more information on each element of the formula or point me to a page explaining them?

 

 

As to why there is gravity well considering GR you'd talk about bending space time

Objects are in a free fall. Massive objects cause curvatures in the space-time continuum and they follow a straight path in this curved space-time. It's correct, isn't it? But what I don't get is: what the free fall thing does really mean? Does it mean that each of the two objects is in fact "falling" toward the other just in the same way a ball falls toward the ground when dropped? However Earth does not fall toward the ball because the ball is not massive enough to curve space-time greatly? Is this what gravity really means?

 

 

Atheist,

the gravitational force on the 3rd object would indeed be zero (forces exerted on an object due to different causes add up)

How so? I don't understand. Isn't it like two persons pulling the same object from opposite directions with the same force? The object cannot move, all right. Is that what you mean by the gravitational force being zero? But the gravitational field of both objects (or the pulling force of the two persons) are still there. It's just that the effect is null because the forces are equal. Did I get it right?

 

Depends on what exactly you are looking at and how you define either of the two. I'm inclined to say "no", but you could possibly explain why you think there could be to get some better replies.

Gravity generates energy (potential energy?). If a massive object causes a gravitational force, it should also be emitting energy. If energy cannot be created or destroyed, only transformed, does the mass of the object remain constant? Do I make any sense?

[Klaynos]

Sure I could, the fastest way is to point you at the wikipedia page...

 

http://en.wikipedia.org/wiki/Law_of_universal_gravitation

 

Yes free fall is how you explain it, and the earth does fall towards the ball, just no where near as fast as the ball to earth...

 

The mass remains constant, the energy is not created, if you throw a ball up it gains gravitational potential energy but it loses kinetic energy at the same rate, when it falls the reverse happens.

[ManOfSteel]

Thanks. Just one more thing. To get back to our three objects, why don't all three fall on each others, collapse and merge into one more massive object? Is it because none of them is massive enough and therefore the curvature of space-time they cause is not enough for any object to pull another completely (to free-fall on another)?

 

The mass remains constant, the energy is not created, if you throw a ball up it gains gravitational potential energy but it loses kinetic energy at the same rate, when it falls the reverse happens.

In the case of the three objects, one object free-falls from its origin toward another object. With the distance between the two decreasing, so does gravitational potential energy. On the other hand, kinetic energy increases at the same rate of the decrease of the gravitational potential energy because the object is accelerated by gravitation while free-falling. So the sum of both energies is null.

[timo]

Thanks. Just one more thing. To get back to our three objects, why don't all three fall on each others, ...

Unless you hold them back somehow or they had some initial velocity (like flying apart from each other), that's exactly what they do.

... collapse and merge into one more massive object?

They could do that. Alternatively, they could bounce off each other, split into tiny pieces flying away, ... .

 

Is it because none of them is massive enough and therefore the curvature of space-time they cause is not enough for any object to pull another completely (to free-fall on another)?

I still refuse to talk about spacetime curvature, partly because the general theory of relativity seems way too advanced for your questions, partly because I don't know the spacetime curvature for that scenario myself (except in the limit of Newtonian gravity ).

According to Newtonian gravity, any mass will create gravitational attraction towards other masses. Use the equations Klaynos and I gave you to calculate the gravitational attraction between two cups of coffee seperated by a distance of 1 meter. Then, assume a frictionless surface (which is btw absolutely not recommended for putting a cup of coffee on) and calculate their acceleration towards each other from a = F/m. You will notice that

a) they do accelerate towards each other

b) the amount of acceleration practically zero, i.e. even on the frictionless surface you wouldn't see any movement within a few days.

 

 

 

In the case of the three objects, one object free-falls from its origin toward another object. With the distance between the two decreasing, so does gravitational potential energy. On the other hand, kinetic energy increases at the same rate of the decrease of the gravitational potential energy because the object is accelerated by gravitation while free-falling. So the sum of both energies is null.

Yes. Hm, wait. I can't just say "yes", I always correct something. ... *looks for something* ... . The sum of the changes is zero, the sum can be any value it had at the start .

[ManOfSteel]

Unless you hold them back somehow or they had some initial velocity (like flying apart from each other), that's exactly what they do.

...

They could do that. Alternatively, they could bounce off each other, split into tiny pieces flying away, ... .

Then why didn't it happen a long time ago with the sun and Earth, for example. What's holding them back? Do they have an initial velocity? Where is this initial velocity coming from then? Is it coming from a centrifugal force? Then where did this centrifugal force come?

[Spyman]

Then why didn't it happen a long time ago with the sun and Earth, for example. What's holding them back? Do they have an initial velocity? Where is this initial velocity coming from then? Is it coming from a centrifugal force? Then where did this centrifugal force come?

The initial nebula had some small amount of net rotation before it collapsed to form our solar system.

 

The Solar System is believed to have formed according to the nebular hypothesis

...

This theory holds that 4.6 billion years ago the Solar System formed from the gravitational collapse of a giant molecular cloud.

...

The region that would become the Solar System, known as the pre-solar nebula, had a diameter of between 7000 and 20,000 AU and a mass just over that of the Sun (by between 0.1 and 0.001 solar masses).

http://en.wikipedia.org/wiki/Solar_system

 

As it collapses, three physical processes shape the nebula: it heats up, its spin increases, and it flattens. The nebula heats up because atoms move more quickly as they fall deeper into the gravitational well and become denser, colliding more frequently: gravitational potential energy is converted to kinetic energy of the atoms, or thermal energy. Second, while initially imperceptible, the solar nebula had some small amount of net rotation (angular momentum), and because angular momentum is conserved, the nebula must rotate more quickly as it shrinks in size. Finally, the nebula must also flatten into a disk, called a protoplanetary disk, as collisions and mergers of blobs of gas average out their motions in favor of the direction of the net angular momentum.

http://en.wikipedia.org/wiki/Planetary_formation

[ManOfSteel]

So they do not collapse on each others under their gravitational forces because a centrifugal force (angular momentum) dating back to the formation of the solar system is holding them back as if they were flying apart from each others. Am I getting it right?

 

Thank you everybody.