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Let's explore the non-inertial frame of reference, of the people in the station, in those both scenarios. All non-inertial forces are the following: $$ \mathbf g = \mathbf\omega\times (\mathbf\omega\times\mathbf r), \quad\quad \mathbf a_c = \mathbf\omega\times\mathbf v $$

Let's explore the non-inertial frame of reference, of the people in the station, in those both scenarios. All non-inertial forces are the following: $$ \mathbf g = \mathbf\omega\times (\mathbf\omega\times\mathbf r), \quad\quad \mathbf a_c = 2\mathbf\omega\times\mathbf v $$

Let's explore the non-inertial frame of reference, of the people in the station, in those both scenarios. All non-inertial forces are the following: $$ \mathbf g = \mathbf\omega\times (\mathbf\omega\times\mathbf r), \quad\quad \mathbf a_c = \mathbf\omega\times\mathbf v $$

Where $\mathbf g$ is the centrifugal gravity acceleration, and $\mathbf a_c$ is the coriolis acceleration, $v$ is the velocity of the ball in this non-inertial reference. Those are vectors. The cross product $\times$ between the vectors returns a vector perpendicular to both simultaneously. Therefore, in a not moving ball with respect to inertial frame of reference, the total non-inertial accelerations are: $$ a_t = 2\omega v - \omega^2 r $$

The direction is radial, pointing to the central axis. Therefore, we can see that centrifugal gravity pushes you away, and Coriolis inwards. If ball is in central axis: $r = 0$ and $v = 0$, no acceleration, will be there. If ball is not in central axis: $a_t = 2\omega v - \omega^2 r$, since this case Coriolis will be bigger, will act like an "centripetal force" preventing it from reaching the ground, and keeping it's movement circular.

I have lots of trouble understanding circicular motion, fictitious forces, and how to dermine if a reference frame is intertial.

What is a reference frame? There are some formal definitions. But, let's pick an intuitive one to boost understanding:

1. Definition: An isolated particle is a particle where the distance of the nearest is infinitly distant away. Therefore, an isolated particle will experience no interaction with another particles, and therefore no force.
2. Definition: An inertial frame of reference is a frame of reference in which an isolated particle is with a constant velocity $v$.

Once the ball leaves your hand, there are no forces acting on it (assume air resistance is negligible). Naively I would think that the ball continues to travel at constant velocity, but this conclusion is likely wrong.

If you are in a reference frame, observing your space station, when someone throws a ball, Newton's second law says: $\mathbf F = \mathbf{\dot p}$. Which means, if there's no force on the ball, linear momentum $\mathbf p = m\mathbf v$ will conserve. Therefore, the ball will travel at a constant velocity, with respect to an inertial frame of reference. However, for those on the station, where their reference frame isn't inertial, they will see the ball curving in it's trajectory, appearing it has a force.

To extend the scenario, imagine that we placed the ball in space, and constructed the space ship around the ball, then begin rotating the space ship. Everyone standing in the spaceship would feel 1g exerted on their feet, and so the gravity simulation would seem to be working well. But then they see the ball floating above their heads, with no forces acting on it, and they would wonder why the simulation does not apply to the ball.

The ball in this case is an inertial reference frame. If you do this, the ball won't fall. Because, your analysis is correct, because you are a inertial frame of reference. The rotating people in the station in this case, is not. On their reference frame, they will see the ball rotating the station while they are not moving. Which means, the ball is accelerating in this case, while they are not moving. Since this force actually doesn't exist (because people are not a valid inertial reference frame), they are called inertial forces. If in a particular example the ball is exactly in the rotation axis of the station, they will see the ball not moving, while they not move. But, the ball also will not fall, because in their reference frame, the "gravity" depends on the radius $r$ from the center to the surface. And, since the ball is exactly in the axis, $r=0$ and no gravity for the ball in this non-inertial frame.

I have lots of trouble understanding circicular motion, fictitious forces, and how to dermine if a reference frame is intertial.

What is a reference frame? There are some formal definitions. But, let's pick an intuitive one to boost understanding:

1. Definition: An isolated particle is a particle where the distance of the nearest particle is infinitely away. Therefore, an isolated particle will experience no interaction with another particles, and therefore no force.
2. Definition: An inertial frame of reference is a frame of reference in which an isolated particle is with a constant velocity $v$.

Once the ball leaves your hand, there are no forces acting on it (assume air resistance is negligible). Naively I would think that the ball continues to travel at constant velocity, but this conclusion is likely wrong.

If you are in a reference frame, observing your space station, when someone throws a ball, Newton's second law says: $\mathbf F = \mathbf{\dot p}$. Which means, if there's no force on the ball, linear momentum $\mathbf p = m\mathbf v$ will conserve. Therefore, the ball will travel at a constant velocity, with respect to an inertial frame of reference. However, for those on the station, where their reference frame isn't inertial, they will see the ball curving in it's trajectory, appearing it has a force.

To extend the scenario, imagine that we placed the ball in space, and constructed the space ship around the ball, then begin rotating the space ship. Everyone standing in the spaceship would feel 1g exerted on their feet, and so the gravity simulation would seem to be working well. But then they see the ball floating above their heads, with no forces acting on it, and they would wonder why the simulation does not apply to the ball.

The ball in this case is an inertial reference frame. If you do this, the ball won't fall. Because, your analysis is correct, because you are a inertial frame of reference. The rotating people in the station in this case, is not. On their reference frame, they will see the ball rotating the station while they are not moving. Which means, the ball is accelerating in this case, while they are not moving. Since this force actually doesn't exist (because people are not a valid inertial reference frame), they are called inertial forces. If in a particular example the ball is exactly in the rotation axis of the station, they will see the ball not moving, while they not move. But, the ball also will not fall, because in their reference frame, the "gravity" depends on the radius $r$ from the center to the surface. And, since the ball is exactly in the axis, $r=0$ and no gravity for the ball in this non-inertial frame.

Just for the record, in the non-inertial reference frame of the people in the rotating space station, the non-inertial acceleration responsible for "gravity" is: $$ g = \omega^2 r $$

Where $\omega$ is the angular speed of the station, and $r$ the distance of a person from the central axis.