It can be shown that for a uniform sphere the force of gravity at a point inside the sphere depends only on the mass closer to the center than that point. The net force of gravity due to points outside the radius of the point cancels.
How far would you have to drill into the Earth in kilometers, to reach a point where your weight is reduced by 4.5% ? Approximate the Earth as a uniform sphere.
^that is so wrong
To find the distance you would need to drill into the Earth to reduce your weight by 4.5%, we can use the concept of gravitational field strength and the fact that the force of gravity inside a uniform sphere only depends on the mass closer to the center.
First, let's derive the expression for the gravitational field strength inside a uniform sphere. The gravitational field strength, g, inside a uniform sphere at a distance r from the center can be calculated using the equation:
g = (G * M(r)) / r^2
where G is the gravitational constant, M(r) is the total mass enclosed within radius r, and r is the distance from the center of the Earth.
Now, let's consider two points: one on the surface of the Earth at radius R and another point at a distance r from the center of the Earth where your weight is reduced by 4.5%.
The force of gravity at the surface of the Earth is given by:
F_surface = (G * M) / R^2
where M is the mass of the Earth.
The force of gravity at the point inside the Earth is given by:
F_inside = (G * M(r)) / r^2
where M(r) is the mass enclosed within radius r.
Since your weight is reduced by 4.5%, we can write an equation using the comparison of these two forces:
F_inside = (1 - 4.5%) * F_surface
Now, we can substitute the expressions for F_surface and F_inside:
(G * M(r)) / r^2 = (1 - 4.5%) * (G * M) / R^2
Simplifying the equation, we get:
M(r) / r^2 = (1 - 4.5%) * M / R^2
M(r) / (1 - 4.5%) = M * (r / R)^2
M(r) = M * (r / R)^2 * (1 - 4.5%)
Canceling out the M, we get:
M(r) = M * (r / R)^2 * 0.955
Now, we can solve for the value of r to find the distance you would need to drill into the Earth:
r / R = sqrt(M(r) / (M * 0.955))
r = R * sqrt(M(r) / (M * 0.955))
Substituting the values of R and M from known data:
R = 6,371 km (approximate radius of the Earth)
M ≈ 5.972 × 10^24 kg (approximate mass of the Earth)
Now, we need to find M(r), which is the mass enclosed within radius r. To do this, we need to determine the density distribution within the Earth.
Assuming the Earth's density distribution is constant up to a certain depth, let's say d km, we can calculate the corresponding mass enclosed within that radius:
M(r) = (4/3) * π * d^3 * ρ
where ρ is the average density of the Earth.
By substituting this new expression for M(r) into the equation for r, we can solve for the value of r.
Therefore, to calculate the distance, you would need to drill into the Earth to reduce your weight by 4.5%, you'll need the average density of the Earth and the density distribution up to a certain depth (d) within the Earth. Using these values, you can calculate r using the equations mentioned above.
Me(r)=Me (r/re)^3 for r<re
so you want g' to be .955g
g= GMe/re^2
g'=.955GMe/re^2 =GMe(r/re)^3
which reduces to..
.955re=r^3 so
r=re *cubrt(.955)
check my work.
you want, of course, the depth of drilling, re-r