3 ms·

The net force acting on it the instant it reaches the very apex is 0, but only for an instant. The force will necessarily increase afterwards, because, through

The net force acting on it the instant it reaches the very apex is 0, but only for an instant. The force will necessarily increase afterwards, because, through its trajectory, the (2nd) derivative of the force is not 0. To put it another way, nothing changes for the particle when it reaches the apex, the net force acting on it is still F(t) = (1/144)(t-T)^2. Just because F(T) happens to be 0 doesn't mean that F(T+dt) has to be 0, it will be (1/144)(dt)², in the same direction as it was before reaching the summit. Because of the above, the direction is also fully determined. It will fall back off on the same trajectory it came up on, if I have my signs straight (if not, then it will continue in the same trajectory on the other side of the dome). Consider a modified version of the car experiment, where instead of an oscillator that is independent of the car, we set up some kind of pendulum pressing on the breaks such that they get activated when the car reaches a certain acceleration, and it gets depressed when the acceleration reaches some negative value. Also, instead of an engine and a flat plane, we can put the car with no engine on a sloped plane. So, the only energy we need to care about is still gravitational. Nevertheless, the car will start and stop continuously, temporarily reaching 0 acceleration and net force and then starting up again.

wat10000 2y ago
“The force will necessarily increase afterwards” What is this force? What causes it to increase?
- tsimionescu 2y ago
  We don't have a term for this, but it's like an inertia for forces acting on the particle (gravity and the surface normal). You can also view this as a missing piece in Newton's laws, as this is not explicitly covered as a possibility. But that is also true then for the entire motion, as all Newtonian equations of motion have the derivatives of acceleration equal to 0, while the motion along this surface necessarily doesn't.
  - wat10000 2y ago
    No such term exists in Newton’s laws. If a particle accelerates, it must be because of a force acting on it. Not something “like an inertia for forces,” but an actual, physical force. Two forces act on this particle, but they sum to zero. If you claim the particle begins to move from zero velocity, then either those two forces don’t sum to zero, or there’s a third force. But they do and there isn’t.
    - tsimionescu 2y ago
      Yes, this "inertia-of-force" is not part of Newton's laws, but it's a straightforward extension of Newton's laws, not a contradiction. And it preserves all of the other interesting properties - energy conservation, time symmetry, determinism. It's certainly less of a contradiction of Netwon's laws than the idea that the particle just sitting there can suddenly start moving, which is a straightforward contradiction of Netwon's first law, and implies that a particle can acquire higher order acceleration (jerk or snap) out of thin air. Of course, this is all moot anyway, as this type of dome with a point-like apex is not a physical system to begin with: since matter is made of atoms, it can't have this type of singular point. Not even going into problems of the precision needed to give the particle just the right initial speed to actually have exactly 0 velocity at the top.
      - wat10000 2y ago
        Inventing a different physics and declaring it to be more correct certainly works, but is kind of a boring answer to the problem. It plainly is a contradiction given that there’s a system where it gives a different answer. If it wasn’t a contradiction then it would be equivalent (or somehow a superset?) and wouldn’t solve the problem. Ignoring the problem as unphysical makes much more sense to me. The real world doesn’t follow Newtonian physics anyway, it’s effectively just a simplification useful for certain common situations. No need to care about edge cases that can’t happen.

wat10000 2y ago

“The force will necessarily increase afterwards” What is this force? What causes it to increase?

tsimionescu 2y ago
We don't have a term for this, but it's like an inertia for forces acting on the particle (gravity and the surface normal). You can also view this as a missing piece in Newton's laws, as this is not explicitly covered as a possibility. But that is also true then for the entire motion, as all Newtonian equations of motion have the derivatives of acceleration equal to 0, while the motion along this surface necessarily doesn't.
- wat10000 2y ago
  No such term exists in Newton’s laws. If a particle accelerates, it must be because of a force acting on it. Not something “like an inertia for forces,” but an actual, physical force. Two forces act on this particle, but they sum to zero. If you claim the particle begins to move from zero velocity, then either those two forces don’t sum to zero, or there’s a third force. But they do and there isn’t.
  - tsimionescu 2y ago
    Yes, this "inertia-of-force" is not part of Newton's laws, but it's a straightforward extension of Newton's laws, not a contradiction. And it preserves all of the other interesting properties - energy conservation, time symmetry, determinism. It's certainly less of a contradiction of Netwon's laws than the idea that the particle just sitting there can suddenly start moving, which is a straightforward contradiction of Netwon's first law, and implies that a particle can acquire higher order acceleration (jerk or snap) out of thin air. Of course, this is all moot anyway, as this type of dome with a point-like apex is not a physical system to begin with: since matter is made of atoms, it can't have this type of singular point. Not even going into problems of the precision needed to give the particle just the right initial speed to actually have exactly 0 velocity at the top.
    - wat10000 2y ago
      Inventing a different physics and declaring it to be more correct certainly works, but is kind of a boring answer to the problem. It plainly is a contradiction given that there’s a system where it gives a different answer. If it wasn’t a contradiction then it would be equivalent (or somehow a superset?) and wouldn’t solve the problem. Ignoring the problem as unphysical makes much more sense to me. The real world doesn’t follow Newtonian physics anyway, it’s effectively just a simplification useful for certain common situations. No need to care about edge cases that can’t happen.