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$T=9$ days

Giancoli 6th Edition, Chapter 5, Problem 65

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Chapter 5, Problem 65 is solved.

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Transcript for this Giancoli solution

We’re going to make a simplifying assumption for this question and say that the effect of gravity from the sun at this large distance is going to be negligible and that instead we’re going to have to have our apparent gravity provided just by the speed of our rotation. So we’re going to have centripetal acceleration which is going to equal to ‘G’ is going to be four times pi squared 'r' over ‘T’ squared. Let’s solve for ‘T’, the period you need to have in order to experience the required centripetal acceleration ‘G’, simplifying and taking the square root of both sides: ‘T’ equals the square root of four times pi squared times 'r' over ‘g’. Substituting in numbers we have: the square root of four times three point one four squared times one hundred and forty nine point six times ten raised to power nine meters all divided by nine point eight newtons per kilogram or meters per second squared which equals seven point seven five nine one times ten raised to the power five seconds. So with that as the orbital period, let’s turn that into something we can imagine. So we have seven point seven five nine one times ten raised to power five seconds times an hour for every three thousand six hundred seconds times a day for every twenty four hours and this gives us eight point nine eight days which we’ll round off to nine days. That’s how long a year would be in order to have a centripetal acceleration due to your speed equal to nine point eight. Why the sun’s gravity is negligible here: you van imagine that the sun’s gravity is really small we’re so far away and the force that it exerts on the earth is huge but that’s only because the earth is really massive. There's the sun and here is the earth, during the day you are on the sun side of the earth and if you were to jump can you imagine it feeling any different compared to at night time when instead the earth moved along in its orbit a little bit more? At night time you’re on the dark side of the earth and if you can imagine jumping here on the opposite side of the earth from the sun so in the day case you have the sun pulling you towards it and in the night case if the sun’s gravity was large and noticeable you would have a harder time jumping at night because it’s pulling you back to the ground at night whereas during the day it’s pulling you away from the earth. So if the sun’s gravity was not negligible you would experience a difference in the day versus the night and we don’t see that so the sun’s gravity is negligible on this kind of scale. The only visible effect the sun’s gravity has on earth has to do with large scale phenomena like ocean tides.