Calculate the distance from the center of the Earth

[envscigrl]

Problem:
Calculate the distance from the center of the Earth to a point below the surface where the acceleration due to gravity is 0.50g, where g is the acceleration due to gravity at the Earth's surface.
I thougtt I could use g = GMe / R^2 solving for R. But this did not work.
This problem seems really simple I just can't seem to figure it out.

 

[Doc Al]

I thougtt I could use g = GMe / R^2 solving for R.

That equation is only good for points at or above the Earth's surface. You need to figure out the force of gravity beneath the surface, for r < Re. Hint: treat the Earth as a sphere of uniform density. The gravity experienced at any given distance from the center r depends only on the mass located beneath that point (the force from the shell of mass above the point cancels out).

 

[wais]

To calculate the distance from the center of the Earth to a point below the surface where the acceleration due to gravity is 0.50g, we need to use the equation for the acceleration due to gravity, which is g = GMe / R^2, where g is the acceleration due to gravity, G is the gravitational constant, Me is the mass of the Earth, and R is the distance from the center of the Earth.

To solve for R, we can rearrange the equation to R = √(GMe / g). However, this equation will only give us the distance from the center of the Earth to the surface, not to a point below the surface where the acceleration due to gravity is 0.50g.

To find the distance to a point below the surface, we need to take into account the fact that the acceleration due to gravity decreases as we move away from the surface. This can be done by using the inverse square law, which states that the force of gravity is inversely proportional to the square of the distance.

So, to find the distance to a point below the surface where the acceleration due to gravity is 0.50g, we can use the equation g = GMe / (R + h)^2, where h is the depth below the surface. We can rearrange this equation to solve for R, which gives us R = √(GMe / 0.50g) - h.

Therefore, to calculate the distance from the center of the Earth to a point below the surface where the acceleration due to gravity is 0.50g, we need to know the mass of the Earth, the gravitational constant, and the depth below the surface. This problem may seem simple, but it involves taking into account the changing acceleration due to gravity as we move away from the surface. I hope this explanation helps you understand the solution better.