When calculating acceleration due to gravity inside the Earth, we need to consider two important facts:
1. As we go deeper into the Earth, the mass of the Earth inside the radius at that depth becomes responsible for gravity, not the whole mass of the Earth.
2. According to the Shell Theorem, when we are inside a spherical shell, the net gravitational force experienced by a point mass is zero.
So, when we are 1000 km below the surface, the mass of the Earth above us (in a shell with a thickness of 1000 km) has no effect on our weight. We only need to consider the mass of the Earth inside the sphere with radius 6371 - 1000 = 5371 km.
Let's denote the total mass of the Earth as M_e and radius as R_e. The mass of Earth inside the smaller radius, R_ins (5371 km), can be called M_ins. We can find M_ins by assuming Earth has uniform density:
Density of Earth = p_e = M_e / ((4/3) * pi * R_e ^ 3)
Now, we can find M_ins with this density and the sphere of radius R_ins:
M_ins = p_e * (4/3) * pi * R_ins ^ 3
Now, we can find acceleration due to gravity at 1000 km below the surface:
a(g)_ins = (GM_ins) / (R_ins ^ 2)
Dividing a(g)_ins by a(g)_surface (or g), we get the ratio:
Ratio = a(g)_ins / g = [(GM_ins) / (R_ins ^ 2)] / [(GM_e) / (R_e ^ 2)] = (M_ins * R_e ^ 2) / (M_e * R_ins ^ 2)
Plugging in the values of M_ins, R_e, and R_ins in the ratio, we can find the actual ratio.
The ratio you are calculating above is slightly off, since you mixed some terms. But if you follow the steps I explained, you should get a correct answer. And the gravity inside the Earth should be less as we go deeper.
What is the ratio of the wight of a man when he is in a tunnel 1000 km below the surface of the Earth to his weight at the surface?
...I'm not sure how to proceed, not because I don't know how to do it, but because I have no idea how to come up with a(g) 1000 km inside the earth. I tried applying the law of shells (or whatever it's called) and:
a(g) = [GM(ins)]/(R^2)
*M(ins) = p*(4/3)*pi*R^3
so:
a(g) = [G * p*(4/3)*pi*R^3]/(R^2)
= 3 G M/(4 * pi * R^2)
...but when I end up plugging that back in, I get a value of a(g) that is like 3.0, which doesn't make any sense since we're closer to the center of the earth, so shouldn't a(g) be larger??
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