4 ms·

The formulation that the forces “cancel out” at the Lagrange points bothers me, even though it’s not technically wrong. At each of the five points, there’s a n

The formulation that the forces “cancel out” at the Lagrange points bothers me, even though it’s not technically wrong. At each of the five points, there’s a net gravitational force towards the sun (to be more precise: towards the common barycenter). This is easiest to see at L2 and L3 where the gravitational forces pull in the same direction. There are now two ways to look at this combined gravitational force: 1. In an inertial reference frame (the one we imagine when we say the Earth revolves around the Sun), this combined gravitational force acts as a centripetal force. At the Lagrange points, it has precisely the correct value to keep the object in an orbit with period 1 year. 2. In the reference frame where Sun and Earth don’t move (that is, the reference frame that rotates at 1/year around the barycenter), every point experiences an apparent centrifugal force. At the Lagrange point, this centrifugal force cancels with the gravitational forces to keep the object stationary in this reference frame. The math is completely the same (x=y vs. x-y=0). I personally just find the first view to be more intuitive.

sreevisakh 5y ago
I occasionally come across this argument that there is no gravity on anything in space (orbit). Their reasoning is that the gravitational force is canceled out by centrifugal force. You could say that the forces cancel out, if you consider pseudo-forces like centrifugal force or Coriolis force as real forces. However, that view sometimes lead to very misleading notions like 'there is no gravity in space'. It also makes explanation of other phenomena like tidal forces needlessly complicated. For this reason alone, I discourage people from analyzing in non-inertial frames. Inertial frames make life simpler and clearer by removing caveats on Newton's second law. You no longer need to invent imaginary forces.
- stouset 5y ago
  I apologize for my ignorance, but how does one consider gravity a "force" from within an inertial reference frame? As I understand it, from the perspective of an observer floating in space inside the gravity well of a nearby object, that observer experiences no net forces. They experience acceleration from the perspective of an outside observer, but not from their own reference frame. A classic example is a person inside a falling elevator. From their own perspective, they're floating with no external forces acting upon them (until, of course, the elevator crashes into the ground). Only when the elevator is held stationary (from the POV of an external reference frame) does the occupant feel an acceleration force upward pushing back against the gravitational well they're standing in.
  - sreevisakh 5y ago
    Your problem of falling elevator is probably the simplest and best example of problem with non-inertial frames. By definition, an inertial frame is a reference frame where Newton's laws of motion can be applied without caveats. For that to be true, the reference frame must be non-accelerating and non-rotating. By that definition, the reference frame attached to the elevator is a non-inertial frame - because it is accelerating towards ground. In your words, the 'perspective of the outside observer' is the actual inertial frame. Now let's see why there is no gravity from the elevator's frame of reference. Remember that by Newton's law of gravitation, gravitational force is GMm/r^2. If you plug in the values, you won't get a zero. This means that the person inside the elevator is definitely facing gravitational 'force'. (I'm assuming classical mechanics where gravity is a force. Relativity considers it as another pseudo force. But that view won't affect our current discussion) Consider the non-inertial frame inside the elevator first. In order for the person to be experiencing no force, all forces on him must be balanced and cancelling out. We know of only gravitational force that is acting downwards. So to cancel it out, we need to invent an equal and opposite force on the person. That force is called 'rectilinear acceleration force' and its value is given by ma (upwards), where m is the mass of the person and a is the elevator's downwards acceleration. So the net force on the person is mg-ma. Since elevator is in free fall, a=g and so the force on the person is zero. You can see the mess here. You have to invent a new force that doesn't exist and have to be careful about its direction too. If the frame is rotating, then more pseudo forces have to be invented - the Coriolis force, Centrifugal force and Euler's force. All these 4 forces don't exist in reality, and are there purely to account for non-inertial frames. Now let's see how the forces play out when considered from the inertial frame of the external observer. Let's consider if the lift is stationary first. Obviously the person inside the elevator is feeling gravitational force that is equal to his weight. In reality, he is not actually sensing gravity acting on him - he is instead sensing the 'reaction force'. The reaction force is the force that the floor of the elevator is applying on him to counteract his weight. In other words, the reaction force from the floor actually cancels his weight - so he remains stationary instead of accelerating towards the ground. Now, unlike pseudo forces, reaction forces are real forces. In this case, it's provided by the electromagnetic repulsion between the molecules on the floor and in the person. This reaction forces don't just act on the person's foot. It acts throughout his body. For example, his internal organs don't fly off under gravity because the reaction force provided by the connective tissues hold them in place, counteracting gravity. Finally what happens when the elevator is falling? The elevator is in free fall, so its acceleration is g towards ground. The person is also falling with the same acceleration g. This means that there is no reaction force acting on him to counteract the gravity. In other words, the person is no longer supported by the floor - that support was the reaction force. Remember that I said that the reaction force was the force that the person was sensing as gravity. Now that reaction force has disappeared. So the person in effect feels that he is weightless, even though gravity is still acting on him. This is the advantage of inertial frames. All Newton's laws can be applied in terms of real and concrete forces with well known causes. You don't need to invent anything new and then justify their existence.
- KolenCh 5y ago
  Non-inertial frame isn’t the problem, proper language is.

sreevisakh 5y ago

I occasionally come across this argument that there is no gravity on anything in space (orbit). Their reasoning is that the gravitational force is canceled out by centrifugal force. You could say that the forces cancel out, if you consider pseudo-forces like centrifugal force or Coriolis force as real forces. However, that view sometimes lead to very misleading notions like 'there is no gravity in space'. It also makes explanation of other phenomena like tidal forces needlessly complicated. For this reason alone, I discourage people from analyzing in non-inertial frames. Inertial frames make life simpler and clearer by removing caveats on Newton's second law. You no longer need to invent imaginary forces.

stouset 5y ago
I apologize for my ignorance, but how does one consider gravity a "force" from within an inertial reference frame? As I understand it, from the perspective of an observer floating in space inside the gravity well of a nearby object, that observer experiences no net forces. They experience acceleration from the perspective of an outside observer, but not from their own reference frame. A classic example is a person inside a falling elevator. From their own perspective, they're floating with no external forces acting upon them (until, of course, the elevator crashes into the ground). Only when the elevator is held stationary (from the POV of an external reference frame) does the occupant feel an acceleration force upward pushing back against the gravitational well they're standing in.
- sreevisakh 5y ago
  Your problem of falling elevator is probably the simplest and best example of problem with non-inertial frames. By definition, an inertial frame is a reference frame where Newton's laws of motion can be applied without caveats. For that to be true, the reference frame must be non-accelerating and non-rotating. By that definition, the reference frame attached to the elevator is a non-inertial frame - because it is accelerating towards ground. In your words, the 'perspective of the outside observer' is the actual inertial frame. Now let's see why there is no gravity from the elevator's frame of reference. Remember that by Newton's law of gravitation, gravitational force is GMm/r^2. If you plug in the values, you won't get a zero. This means that the person inside the elevator is definitely facing gravitational 'force'. (I'm assuming classical mechanics where gravity is a force. Relativity considers it as another pseudo force. But that view won't affect our current discussion) Consider the non-inertial frame inside the elevator first. In order for the person to be experiencing no force, all forces on him must be balanced and cancelling out. We know of only gravitational force that is acting downwards. So to cancel it out, we need to invent an equal and opposite force on the person. That force is called 'rectilinear acceleration force' and its value is given by ma (upwards), where m is the mass of the person and a is the elevator's downwards acceleration. So the net force on the person is mg-ma. Since elevator is in free fall, a=g and so the force on the person is zero. You can see the mess here. You have to invent a new force that doesn't exist and have to be careful about its direction too. If the frame is rotating, then more pseudo forces have to be invented - the Coriolis force, Centrifugal force and Euler's force. All these 4 forces don't exist in reality, and are there purely to account for non-inertial frames. Now let's see how the forces play out when considered from the inertial frame of the external observer. Let's consider if the lift is stationary first. Obviously the person inside the elevator is feeling gravitational force that is equal to his weight. In reality, he is not actually sensing gravity acting on him - he is instead sensing the 'reaction force'. The reaction force is the force that the floor of the elevator is applying on him to counteract his weight. In other words, the reaction force from the floor actually cancels his weight - so he remains stationary instead of accelerating towards the ground. Now, unlike pseudo forces, reaction forces are real forces. In this case, it's provided by the electromagnetic repulsion between the molecules on the floor and in the person. This reaction forces don't just act on the person's foot. It acts throughout his body. For example, his internal organs don't fly off under gravity because the reaction force provided by the connective tissues hold them in place, counteracting gravity. Finally what happens when the elevator is falling? The elevator is in free fall, so its acceleration is g towards ground. The person is also falling with the same acceleration g. This means that there is no reaction force acting on him to counteract the gravity. In other words, the person is no longer supported by the floor - that support was the reaction force. Remember that I said that the reaction force was the force that the person was sensing as gravity. Now that reaction force has disappeared. So the person in effect feels that he is weightless, even though gravity is still acting on him. This is the advantage of inertial frames. All Newton's laws can be applied in terms of real and concrete forces with well known causes. You don't need to invent anything new and then justify their existence.
KolenCh 5y ago
Non-inertial frame isn’t the problem, proper language is.