Does the Coriolis force apply to an object moving weightlessly in a tunnel around the center of Earth? I just want to be sure I understand correctly.
They ask us to find the speed at which an object moves weightlessly in a tunnel around the center of the Earth. Assume Earth is homogeneous, and the part of Earth beyond the tunnel does not contribute to gravity.
I suppose the gravity and centrifugal force need to cancel each other. But I was wondering if we also need to take the Coriolis force (or any other force by the way) into account?
If the object moves weightlessly in the tunnel, is it as if the object was moving in orbit around Earth? Because it doesn't touch the walls of the tunnel, shouldn't the Coriolis force apply?
The Coriolis force only applies to an object that was in contact with Earth, and then not any more. But in this case, if the object doesn't touch the wall, this doesn't apply, do I understand this correctly?
 A: 
But I was wondering if we also need to take the Coriolis Force (or any other force by the way) into account ?
If the object moves weightlessly in the tunnel, it's as if the object was moving in orbit around Earth ? Because it doesn't touch the walls of the tunnel, the Coriolis Force shoudn't apply ?

The Coriolis force is a fictitious force, meaning that it appears whenever we are doing our analysis of physics using a rotating reference frame. In that rotating reference frame it applies for all objects, regardless of whether they are in contact with the Earth or not.
This force is not due to contact with the Earth, it is due to the fact that we are using a non-inertial coordinate system to do our analysis. So everything in that analysis is affected. As a hint for your specific problem, it may be that you can build your tunnels in a specific location which will eliminate or simplify the Coriolis force.
A: If your tunnel goes south to north, you can not avoid to bump to the wall, since at the equator your initial velocity perpendicular to the tunnel is that of the earth rotating. Farther north, the velocity of the tunnel is slower, so you will bump into it, and this you could attribute to a Coriolis force in the rotating system.
If your tunnel goes around the equator you can have this weightless path.
A: Coriolis force with a horizontal component occurs if (i) you are working in a frame of reference that is rotating with the Earth's surface and (ii) the object's motion has any north-south component.
Since the tunnel is attached to the Earth's surface, condition (i) applies. Unless the tunnel is constructed underneath the equator then condition (ii) applies.
This is why the groundtrack of a satellite such as the ISS does not close after one orbit. In space it returns to the same position after one orbit, but its groundtrack (projected onto the surface of the Earth) is not closed because the Earth has rotated underneath it. In the ground reference frame, the satellite experiences a sideways force which is the Coriolis force. Your tunnel scenario is exactly the same except at a negative altitude.
