Force of gravity if the Earth's radius, r = r/2

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Homework Help Overview

The problem involves calculating the gravitational acceleration at the surface of a hypothetical planet that retains Earth's mass but has half its radius. Participants are exploring the implications of this scenario on the value of gravitational acceleration, denoted as g.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to apply the formula for gravitational acceleration but questions their calculations due to discrepancies with expected results. Other participants suggest checking unit consistency and highlight a simpler conceptual approach based on the relationship between radius and gravitational acceleration.

Discussion Status

Participants are actively discussing the calculations and potential errors. Some have provided guidance on checking units and recognizing the mathematical relationship that leads to the conclusion that the new value of g should be four times the original. There is an acknowledgment of a misunderstanding regarding the units of radius.

Contextual Notes

There is a noted confusion regarding the units used for the radius of the Earth, with participants clarifying whether it should be in meters or kilometers. This has implications for the calculations being discussed.

prace
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Problem: Suppose that the Earth retained its present mass but was somehow compressed to half its present radius. What would be the value of g at the surface of this new, compact planet?

My work: So, this seems pretty simple, and I get the right answer, but I seem to be off by a lot of decimal places. Can anyone tell me what is wrong here with my calculations?

g = (GMe)/(Re)² where Me = mass of the Earth, and Re = radius of the Earth.

So if Re is compressed to half its present radius, then:

g = (GMe)/(Re/2)² = (6.67E-11*5.98E24)/(6370/2)² = 39344273 m/s²

The answer in my text gives 39.2 m/s².

To try and check what I was doing wrong, I tried to calculate for the known value of g = 9.81 m/s² and I got 9836068.3 m/s². So it looks as if the correct numbers are there, but I am somehow messing this up.

Thanks for any help!
 
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Check the units of the quantities that you're calculating with, and make sure that they're consistent with the units of the answer that you're supposed to get.
 
A simpler way to do this, without going through all the numbers, is to notice that the new value of g will be 4 times the original value. (the 2^2 goes to the numerator and 4x9.8 = 39.2 ms^-2)

The only problem with your calculation could be that you've not written down the radius in metres.
 
Oh my... so simple huh?! ^_^ Thank you. It looks like I calculated the radius of the Earth in meters when it should have been kilometers! Thanks for seeing that for me.
 
neutrino said:
A simpler way to do this, without going through all the numbers, is to notice that the new value of g will be 4 times the original value. (the 2^2 goes to the numerator and 4x9.8 = 39.2 ms^-2)

The only problem with your calculation could be that you've not written down the radius in metres.

Oh cool, thanks neutrino, that way is much easier. Thanks!
 
prace said:
It looks like I calculated the radius of the Earth in meters when it should have been kilometers!
No, it's the other way. :smile:
 

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