I was recently reading some Newtonian dynamics textbook, and then I came across with a problem about the centrifugal effect on mass free falling to Earth. I can mathematically appreciate the fact that due to the rotation of the Earth, in the rotating frame, fictitious forces actually reduces the effective gravity. But when I look at it from an inertial frame, I cannot intuitively understand how does the spinning of the Earth makes the mass free falling more slowly than when the Earth is not rotating?
I think the answer must be either the book has got it wrong or that there is some confusion between what the author meant to put in the book and the question before us.
If I understand the question correct the point is does an object falling towards a planet undergravity have different acceleration (different dynamics) depending on whether the earth is rotating or not.
I think the answer is that if the earth is rotating on its axis then there is no difference between the object falling whether or not the earth is rotating - it will of course depend on whether the object itself is rotating around the planet as noted in the question.
Now - on the other hand, if the earth is rotating around the sun then depending on where the object is with respect to the rotation around the sun the dynamics may be effected by centrifugal forces - but then the object would not be in an inertial reference frame
But when I look at it from an inertial frame, I cannot intuitively understand how does the spinning of the Earth makes the mass free falling more slowly than when the Earth is not rotating?
In the inertial frame, the mass will have the same radial acceleration whether rotating or not.
But on a rotating earth, the mass also has a tangential speed. That speed carries it sideways along the curvature of the earth. This has two effects:
- When considering the initial acceleration vector, the ground is no longer at the same distance. It's a little farther away. So impact with the ground takes longer.
- The radial acceleration over time is not in the same direction. The summation of these vectors over time is less than it would be if they were all in the same direction. So the net acceleration over time is less than it would be from an object that had no tangential speed.