# What forces are required to physically separate two bodies in a gravitational system?

I have a logical misconception about resultant force/potential energy that has been bugging me for a while.

Consider lifting a book with a constant force against gravity. I think I understand that when we consider the system as only the book, no resultant force acts on the book and thus no resultant work is done on the book (no change in KE). However, when we consider the system as the book+earth we can begin to include potential energy gained in the system by lifting (with the gravity forces being internal).

Here is my issue though: when providing a single external force (say equal to the weight of the book) on the book (in the book+earth system), does this force separate the two bodies or does it simply accelerate the centre of mass of the system? I read another answer on the site that said, in fact, two equal and opposite forces are required to separate two bodies and increase the potential energy (it is just that, as W = F.x, the displacement of the earth is negligible and we consider the work done to be zero). The precise quote was:

"The only external force is F and let this force have the same magnitude as the weight of the body. That single force F cannot move the body further away from the Earth. To separate the body and the Earth two forces of magnitude F but opposite in direction must act on the body and the Earth."

This seems to make sense to me as, when we lift a book, we are actually providing two forces, the weight of the book is transferred through our body and this provides the increased opposite contact force to push against the earth (though this may be incorrect).

However, if you always need two equal and opposite forces on each body in the system to separate them, how does a rocket leave the earth (from the ground and once in space)? It does not "push" against the earth yet in my mind I consider its thrust to be an single external force on the rocket+earth system which, by the previous statement, should not separate the two bodies in the system, but rather just accelerate the entire earth+body system.

Where is my logical misunderstanding? My apologies if this is too long-winded but thanks for your time!

On one hand, your quote is incorrect. On the other hand, there are always two equal and opposite forces.

Consider a refrigerator magnet and a refrigerator sitting on frictionless ice. If you pull gently on the magnet, you will not separate the magnet from the refrigerator. You will pull the refrigerator and magnet towards yourself. Likewise, you will pull yourself towards the refrigerator and magnet.

If you yank hard on the magnet, you will pull it off. The force of attention between the refrigerator and magnet will rapidly shrink to approximately 0. You will accelerate the magnet rapidly toward yourself, and yourself a little towards the magnet. There will always be a small attraction between the refrigerator and magnet, so the refrigerator will have a small acceleration. But you will most likely approximate it as 0.

• Okay, this does make sense, but is there any difference between the forces in each scenario other than the second one being much larger? Also, could you explain why the quote is wrong and why you can't apply the force to the centre of mass of the whole system? Here is where the original quote was found:(physics.stackexchange.com/questions/16329/…) Nov 8, 2019 at 13:31
• The quote says you can't separate a body and the Earth with just one external force. The two examples are much the same. The exhaust gas from the rocket plays the role of you in the magnet example.you can apply an external force to the center of mass. Just pull gently on the magnet. Nov 8, 2019 at 13:55
• Okay, so when the external force applied on the magnet is larger than the magnetic force of attention from the fridge you get the separation, otherwise you are accelerating them as a system? Nov 9, 2019 at 12:43
• By a system, I mean accelerate them together, attached still. And if you became part of the system and pulled the magnet (i.e. you stood on the frictionless ice, also) then you would accelerate the magnet rapidly toward yourself, and yourself a little towards the magnet, as you said. Nov 9, 2019 at 13:02
• You are free to choose the system as you like - the magnet by itself, the refrigerator plus magnet, refrigerator plus magnet plus you. Usually you stick to the choice once you make it. The difference is what do you consider to be an external force. Momentum is conserved for a system without external forces (isolated system). Just the magnet is accelerated by an external force. That system gains momentum. All forces are internal for magnet + refrigerator + you. The center of mass travels in a straight line at uniform velocity. Nov 9, 2019 at 17:59

However, if you always need two equal and opposite forces on each body in the system to separate them, how does a rocket leave the earth (from the ground and once in space)? It does not "push" against the earth yet in my mind I consider its thrust to be an single external force on the rocket+earth system which, by the previous statement, should not separate the two bodies in the system, but rather just accelerate the entire earth+body system.

Newton's third law states for every force there is an equal and opposite force. The important thing to keep in mind is that the two forces are acting on different objects.

The rocket pushes gases (or liquid), which are really tiny particles of mass, from inside in one direction (toward the earth) with a force that is equal and opposite to the force (thrust) that the gases push on the rocket (away from the earth).

From the perspective of the rocket, the external forces acting on it are the force of the gas acting upward and the force of gravity and air drag acting downward. As long as the force of the gas on the rocket pushing it upward (thrust) is greater than the forces downward (gravity + air drag), the rocket will continue to accelerate upward.

Thanks for your answer! My query is more focused on what is happening in the system of the rocket and the planet. In this system, we can say that the gravity acts as an internal force and the rocket's thrust is an external force on the rocket+earth system. Clearly this force does separate the earth and the rocket, but the statement I quoted in my original question seems to say that it shouldn't (since there is only one force). Why is this? –

I would consider the rocket fuel as part of the rocket. Then all the forces associated with the rocket+earth system are internal, that is, the system is experiencing no external forces. The rocket exerts a force on the earth and the earth exerts an equal and opposite force on the rocket. Both forces are internal to the system.

For conservation of momentum, the momentum of the rocket+earth system after firing the rocket must be the same as before firing the rocket because there are no external forces acting on the system. So while the rocket does move away from the earth, the earth also moves away from the rocket. But the center of mass of the rocket+earth system does not move.

So imagining the basic system of earth+rocket (no friction/atmosphere), when the rocket has fired there is no change in the centre of mass of the rocket (since only internal forces are applied) and therefore there is also no change in the centre of mass of the earth+rocket system.

No, there is movement in the center of mass of the rocket. And there is movement of the center of mass of the earth as well. But there is no movement of the center of mass of the combination of the two. For conservation of momentum (no change in momentum) of the isolated rocket-earth system, the momentum of the rocket and earth have to be equal and opposite, or

$$mV=-Mv$$

Where $$V$$ and $$m$$ are the velocity and mass of the rocket and $$M$$ and $$v$$ are the mass of the earth and velocity at any instant. Since the mass of the earth is so much greater, its velocity is infinitely small and unobservable.

Does this explain the lifting of the book too: when we lift the book, we are applying equal an opposite internal forces on the system so the book moves away from the earth and the earth moves from the book but the centre of mass of both is still?

Correct. Provided you consider your arm, the book, and the earth to be an isolated system, like the rocket and earth, there will be no movement of the center of mass of the combination. Though the earth is technically moving away from the book its movement is infinitesimally small so as not to be unobservable.

Hope this helps.

• Thanks for your answer! My query is more focused on what is happening in the system of the rocket and the planet. In this system, we can say that the gravity acts as an internal force and the rocket's thrust is an external force on the rocket+earth system. Clearly this force does separate the earth and the rocket, but the statement I quoted in my original question seems to say that it shouldn't (since there is only one force). Why is this? Nov 8, 2019 at 13:16
• @user246795 See my update. Nov 8, 2019 at 13:49
• This helped! So imagining the basic system of earth+rocket (no friction/atmosphere), when the rocket has fired there is no change in the centre of mass of the rocket (since only internal forces are applied) and therefore there is also no change in the centre of mass of the earth+rocket system. Does this explain the lifting of the book too: when we lift the book, we are applying equal an opposite internal forces on the system so the book moves away from the earth and the earth moves from the book but the centre of mass of both is still? Nov 9, 2019 at 12:56
• @user246795 It's too long to respond under comments. So I have responded with an update of my answer. I'm sorry I have no more time to spend on this, so I hope it acceptably answers you questions. Nov 9, 2019 at 14:44