Let's suppose we have a penduluim inside a wagon.
When we rotate the wagon, or slide it, the result will be the same : all molecules inside will move with the wagon. As well as when you are inside a train, every movement of the train is translated to you, and to every molecule in the train. If not you would be throwed on the back wall when there is a higher speed. Rotating the train is the same as putting the pendulum inside a building and the earth is rotating the building. So rotating the wagon creates a rotational force that affects everything inside the wagon. You need to understand that before we continue.
We do understand that. We also understand there isn't some magic force that causes everything to rotate.
If the train changes speed, you feel it. You do get thrown back onto a wall.
What stops that is what is connecting you to the train.
Typically that will be your feet or but and back.
This allows a significant transfer of force.
If something is in the air, it doesn't have that.
Instead it relies upon the air.
If the air was going to be strong enough to keep the pendulum rotating, then it wouldn't be able to oscillate.
It would just drop to vertical.
The air is not strong enough to keep the pendulum stationary, and instead allows it to swing back and forth. This means it isn't going to be able to impart enough energy/momentum to keep it moving with the container.
Also, it wouldn't be a rotational force, but I'll get to that later.
it is the same with gravity or not, that force is always present
As well as other forces, or instead you could generalise this "rotational force" to be one of air resistance.
This will be proportional to velocity and act in a direction opposite the velocity of the object (with this velocity being w.r.t the air).
We all know that gravity is what oscillates the pendulum : the ball is attracted to the ground and the momentum brings the ball on the other side, etc, until there is no momentum at all, and the penudulm then stands perfectly still and vertical.
And what stops it moving? The air resistance.
So do you notice how when the air has transferred enough momentum/energy to keep it stationary, it is stationary?
At each time t, there are two forces : gravity (g->) and the rotational force (r->), and they both affect the pendulum
the resulting force r-> + g-> is no equal to g->. Even when it's not "gravity-less", the ball is affected by the rotation as we saw earlier.
And this force r-> is tiny.
Like I said, it should be ar-> (air resistance).
The rotational force affects the extremity of the pendulum (the ball) much more when the ball is at its change of direction : during that time, the ball is almost gravity-less, so we are in the previous phase where we can account that there is no gravity for some time, and that the gravitational force don't counter the rotational force. So at this moment the rotational force is the only force that exist. Because the pendulum is slower at these extremities, forces applied at that moment have greater effect than at the vertical for instance.
No. The forces have the same effect regardless of where they are applied. The exception would be forces due to things like air resistance which are dependent on speed. In that case as the pendulum is the slowest here, the force will be least here.
The force from air resistance is greatest when the string is vertical and the pendulum is moving the fastest. This force acts to slow the pendulum down. If it doesn't do it here, what makes you think it is going to keep it rotating with Earth when it is so much smaller at the extremities.
And again, this isn't a rotational force. It is a linear force. Lets ignore the air resistance component due to its swinging for now (even though that alone shows that this force isn't going to significantly effect it)
So, at the pole, it is at the peak of its swing, and it is forced to the right in your diagram.
It was being pushed to the right from when it moved from vertical to this side. It keeps getting pushed right until it goes back to vertical. The force is also proportional to how far away it is from the centre of rotation.
Now the air is moving the opposite direction, and it is forced left.
This acts to counter the force which was forcing it right.
This means the force on one side counters the force on the other.
This means in total, there is no force trying to make it rotate.
so the conclusion is : there is no way the pendulum would make a 360° rotation on the north pole, it would be way WAY less than that due to the always forgotten rotational force.
Nope. The conclusion is there is no "rotational force".
You just have the force of the air resistance, which is negligible, or else the pendulum would just sit vertically not swinging. This means that if it was going to cause a rotation of the pendulum, it would be insignificant and thus you would likely observe the 360 degree rotation.
But more importantly, it isn't a rotational force. It is a linear force where the 2 sides cancel each other out.
Also, if you wish to assert it would be WAY less, then do the math and show just how much less, because right now, you are like those that say we couldn't be spinning because we would fly off, ignoring that the force of gravity is roughly 300 times stronger, meaning we would still fall to Earth, or for a less "controversial" idea, you couldn't possibly put your foot on the accelerator to speed up a car because you would travel way too fast instantly due to that acceleration which would kill you.
So to conclude, you would expect it to make an apparent 360 degree rotation, as there is no rotational force, and even if there was, it would be insignificant.