Gravitational Force on 10.0kg Mass at Earth Locations

[azila]

Homework Statement

Find the gravitational force that the Earth exerts on a 10.0 kg mass if it is placed at the following locations. Location A: at the surface of the earth; Location B: at the outer surface of the molten outer core; Location C: at the surface of the solid inner core; Location D: at the center of the earth

Homework Equations

p = Me/Ve
Ve = (4/3)pi(Re)^3
F = GMeM/Re^2

The Attempt at a Solution

Ok, so first I found the densities at the four different locations on the earth. So here they are:
Location A: 2000 kg/m^3
Location B: 3300 kg/m^3
Location C: 5500 kg/m^3
Location D: 13000 kg/m^3

To find the gravitational force, I thought I first have to solve for the volume of the Earth and then find the certain radius at that location. So for instance, for location A,
I plugged the rho(density) = mass (which I know as 5.97 * 10^24) all over V (which I don't know); I found V to be 2.985 * 10^21.

Then I took that and plugged it into the Ve equation, and found the radius to be 8.93 * 10^6.

Then I used this radius in the gravitational force equation, where it was
G*10kg*5.97*10^24 allover (the radius squared). I got an answer like 49.9 N, but when I submit the answer, the software tells me that it is wrong. So, what am I doing wrong? Please help!

Thanks.

 

[learningphysics]

What exactly are you given in the question... post it exactly as it is written.

 

[azila]

this is exactly what was written in the problem. The problem tells to look at this diagram of the Earth where four different densities are listed according to the location. The densities that I gave you are listed. So.....

 

[learningphysics]

this is exactly what was written in the problem. The problem tells to look at this diagram of the Earth where four different densities are listed according to the location. The densities that I gave you are listed. So.....

Do they give the radius of location A, B etc... ?

 

[azila]

no, see that's what I am telling you. I first had to find the volume of the Earth at that certain location. The equation is average density = mass of earth/volume of earth.
After using that, I found the radius of the Earth at the different locations by using the volume equation which equals 4/3(pi)Radius^3 and plugged that into the gravitational force equation. However, the answer I get is not the answer.

 

[learningphysics]

no, see that's what I am telling you. I first had to find the volume of the Earth at that certain location. The equation is average density = mass of earth/volume of earth.
After using that, I found the radius of the Earth at the different locations by using the volume equation which equals 4/3(pi)Radius^3 and plugged that into the gravitational force equation. However, the answer I get is not the answer.

Something's strange about this question... 2000kg/m^3 is not the average density of the entire earth... I believe what they are saying is that it is the density of a particular layer of the earth... I'm surprised they aren't giving the radius, or something other information...

You can look up the radius of the earth. It is 6,356.750 km. You can use that to get the force at the position A.

Did they give you the mass of the earth? Or did you have to look that up yourself?

 

[learningphysics]

I think the question expects you to look up the radius' of the different locations... then calculate the mass contained within a sphere of radius A (radius of the earth), B, C or D(which is 0)

using those masses, and the radius' you can calculate the force on a mass at those locations.

 

[learningphysics]

radius of outer core is approx. 3400km
radius of inner core is approx. 1220km

 

[azila]

you know what, I think what you said is right. I just need to have the radius' of the locations and use them in the equation. Thanks so much. I just made it to complicated. I put in the answer for location D, it was right. So, i'll see if the others work.

 

[learningphysics]

you know what, I think what you said is right. I just need to have the radius' of the locations and use them in the equation. Thanks so much. I just made it to complicated. I put in the answer for location D, it was right. So, i'll see if the others work.

Cool... be careful when calculating the masses... for the sphere at location B, you'll need to subtract the mass of the outer layer with density 2000kg/m^3 from the mass of the earth.

Then for the sphere at location C, you'll need to subtract the mass of the next layer from the mass you got for B... etc... hope this makes sense...

 

[azila]

when i calculate for location B, how do I calculate the mass to subtract from the mass of the whole earth? Like what would I times it by? I tried to go through the average density = mass of earth/volume equation but the number I get is volume. Thanks again for helping.

 

[learningphysics]

when i calculate for location B, how do I calculate the mass to subtract from the mass of the whole earth? Like what would I times it by? I tried to go through the average density = mass of earth/volume equation but the number I get is volume. Thanks again for helping.

You don't calculate the location B... you should look it up... I found 3400km.

What you'd do to get the mass is, first find the volume of the outer layer... Take the volume of the outer sphere with the radius of the earth... subtract the volume of the inner sphere... R = 3400km... that's the volume of the layer... then multiply by the density 2000kg/m^3. That gives the mass to be subtracted.

 

[azila]

ok, so this is what i did;
1. the volume of the outer layer with the radius of the earth, is 1.09 *10^21.
2. the volume of the inner layer with radius 3.4 m *10^6, is 1.64636 * 10^20
3. then i subtracted the inner layer volume from the outer layer volume, and got 9.23 * 10^20
4. then I times it by 2000kg/m^3 and the mass is 1.846 * 10^24
5. then i subtracted this mass from the mass of the earth, and got 4.12* 10^24
6. then i plugged this into the gravitational force equation in which I used the radius 3400 km, I got 237.93, which is not the answer. So......Am I not following the directions correctly. Thanks for the step by step process but I don't know what I am doing wrong.

 

[learningphysics]

Your procedure was fine. It's possible the numbers I gave for the inner core and outer core radius are not right. I got them from wikipedia.

Also, I'm getting 238 point something when I do the calculations...

wish the problem had given the the radii...

 

[azila]

yea, I know. But thanks for helping out. The whole procedure you gave made sense. Thank you so much.

 

[learningphysics]

yea, I know. But thanks for helping out. The whole procedure you gave made sense. Thank you so much.

no prob. you're welcome.