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

In summary, the conversation discusses the calculation of the value of g at the surface of a compressed Earth with half its present radius. The correct answer is 39.2 m/s², but the individual's calculations were off by several decimal places. The conversation also mentions checking the units and using a simpler method to calculate the answer. Ultimately, it is discovered that the individual had calculated the Earth's radius in meters instead of kilometers.
  • #1
prace
102
0
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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  • #2
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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!
 
  • #6
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:
 

1. What is the formula for calculating the force of gravity on Earth?

The formula for calculating the force of gravity on Earth is F = G(m1m2)/r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

2. How does the force of gravity change if the Earth's radius is halved?

If the Earth's radius is halved, the force of gravity will decrease by a factor of four. This is because the force of gravity is inversely proportional to the square of the distance between two objects. In this case, the distance between an object on the Earth's surface and its center will be halved, resulting in a decrease of the force of gravity by four times.

3. Will the force of gravity on Earth change if the Earth's mass remains the same but its radius is halved?

Yes, the force of gravity on Earth will change if the Earth's mass remains the same but its radius is halved. As mentioned before, the force of gravity is directly proportional to the masses of the two objects. Therefore, if the distance between them is halved, the force of gravity will also be halved.

4. How does the force of gravity on Earth compare to other celestial bodies?

The force of gravity on Earth is dependent on its mass and radius. Compared to other celestial bodies, such as the sun or other planets, the force of gravity on Earth is relatively weaker. For example, the force of gravity on Jupiter is about 2.5 times stronger than that on Earth.

5. Is the force of gravity on Earth constant?

No, the force of gravity on Earth is not constant. It varies depending on the distance between two objects and their masses. Additionally, factors such as the Earth's rotation, shape, and composition can also affect the force of gravity on different parts of the Earth's surface.

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