Gravity and Radius of the earth problem

[J.Sterling47]

Homework Statement

Hey guys, I have a problem where we measured the gravity on separate floors of a building using a gravometer. It gave us values in mgals. So each floor has a distance of 0.5m. How do we take this into account as we move up and down the floors? As we move up, there's and increase of mass below us and vice versa as we go down. The goal is to calculate the radius of the Earth using this method.

Homework Equations

dg/g = -2(dr/r) where dg is the change in gravity per floor and dr is the height of each floor. We got the radius to be close at about 6500km, but without taking into account the changing of the mass below.

The Attempt at a Solution

Don't really know where to start

 

[DaveC426913]

So each floor has a distance of 0.5m.

You sure about that?

 

[J.Sterling47]

You sure about that?

Yeah sorry it was supposed to be 5m. I converted it wrong but yea

 

[haruspex]

We got the radius to be close at about 6500km, but without taking into account the changing of the mass below.

If you are concerned about that, the first thing is to get an upper bound on how much difference that will make. Estimate the mass per floor of the building. Be generous. Likewise, maximise the effect by assuming all that extra mass is only one floor below (for simplicity). How much difference will that make to your answer?

 

[HalfLostAndSearching]

, any help would be appreciated

I would suggest approaching this problem by first understanding the concept of gravitational acceleration and its relationship to mass and distance. Gravitational acceleration is the force of gravity acting on an object, and it is directly proportional to the mass of the object and inversely proportional to the square of the distance between the objects.

In this case, you are measuring the change in gravitational acceleration (dg) and the change in distance (dr) as you move up and down the floors of a building. To take into account the changing mass below, you can use the equation for gravitational acceleration (g) and rearrange it to solve for mass (M).

g = G(M/r^2)

Where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the surface. By rearranging this equation, we can solve for M:

M = g(r^2)/G

Now, we can substitute this value for M into the equation for dg/g:

dg/g = -2(dr/r)

And rearrange to solve for r:

r = -dg/(2g)(dr)

By plugging in the values for dg and dr that you measured on each floor, you can calculate the radius of the Earth at that specific distance from the center. However, keep in mind that the radius of the Earth will vary at different distances from the center, so you may need to take multiple measurements at different heights to get a more accurate estimate of the Earth's radius.

Additionally, it is important to note that there may be other factors at play that could affect your measurements, such as the building's structure and composition, as well as the precision and accuracy of your equipment. It may be helpful to consult with a physics or geology expert to ensure the accuracy of your calculations.