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?
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$\begingroup$ Related: Centrifugal force? $\endgroup$– GonencMay 8, 2015 at 20:48
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$\begingroup$ @gonenc - no don't think that is related 100% - the question here is about why if the mass is not rotating it should feel any centrifugal force because the earth is rotating. - the point is made that if the mass was in a rotating frame it would make sense - the question is about a non-rotating inertial frame --- and 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... $\endgroup$– tomMay 8, 2015 at 21:04
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1$\begingroup$ @wanwuwi - can you clarify if your question is answered by physics.stackexchange.com/q/108925 $\endgroup$– tomMay 8, 2015 at 21:33
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$\begingroup$ The fact that an observer in an accelerated frame sees a different dynamic than an observer in a rest frame doesn't make any difference to the actual physics of the observed object. More importantly, the accelerated observer can easily tell that he or she is undergoing acceleration and correct the measurements accordingly (which is an important aspect in engineering of control systems, but that's not physics that belongs here). $\endgroup$– CuriousOneMay 8, 2015 at 21:42
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2 Answers
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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.
answered May 12, 2015 at 23:21
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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
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$\begingroup$ What I understand from the question is the following (using a carousel instead of the earth): OP is saying I put a stone on the carousel and I am riding along with it and the stone gets drifted away. Now I am standing outside of the carousel and the stone still get drifted away. Where does this force come from? $\endgroup$– GonencMay 8, 2015 at 21:20
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$\begingroup$ @gonenc - ah ok I see your point of view. - we need clarification from the person who asked the question. I am happy to delete my answer if I have got it wrong, and the chance are that I have.... $\endgroup$– tomMay 8, 2015 at 21:32
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$\begingroup$ In fact, the question I really want to know is that would the particle take longer time to reach the earth if the earth is rotating faster? I cannot understand why would it do so though. $\endgroup$– wanwuwiMay 9, 2015 at 22:20
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$\begingroup$ @wanwuwi the particle would only take longer if it was rotating itself around the earth. $\endgroup$– tomMay 9, 2015 at 23:22
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$\begingroup$ Just another question. If, say the earth has a negligible mass, does that mean as the earth rotates, the particle would appear to be 'pushed away' by the centrifugal force if viewed from a rotating frame? What does that mean in an inertial frame then? $\endgroup$– wanwuwiMay 10, 2015 at 11:16