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, to reach a point where your weight is reduced by 5.5 ? Approximate the Earth as a uniform sphere.
Express your answer using two significant figures.
Reduced by 5.5 m/s^2 (from 9.8 m/s^2), or by a factor of 5.5?
When you have clarified how much you want to reduce g, drill to a value of R for which M/R^2 (which is proportional to density*R) is reduced by the same factor.
No by 5.5%.
To reduce the value of g by 5.5%, reduce the distance from the center of the Earth by 5.5%. That will mean going down about 350 km.
To find the distance you would have to drill into the Earth to have your weight reduced by 5.5, we can use the principle that the force of gravity inside a uniform sphere only depends on the mass closer to the center.
Let's assume that the radius of the Earth is R. As we drill into the Earth, we move closer to the center of the Earth, where the mass above us cancels out.
To determine how much mass is above the point where your weight is reduced by 5.5, we need to find the radius at which this occurs. Let's call this distance x.
Since the force of gravity inside a sphere only depends on the mass closer to the center, we can say that the remaining mass above the point x is responsible for a force that reduces your weight by 5.5.
Now, let's write an equation to represent this situation. The force of gravity on you at the surface of the Earth is equal to your weight, which we'll denote as W:
W = (G * m * M) / R^2 --------------(1)
where
G is the gravitational constant,
m is your mass,
M is the mass of the Earth, and
R is the radius of the Earth.
Next, we need to find the force of gravity when your weight is reduced by 5.5:
W - 5.5 = (G * m * Mx) / (R - x)^2 --------------(2)
where
Mx is the mass that is closer to the center of the Earth than the point x.
Now, we can solve equation (2) for Mx:
Mx = (W - 5.5) * (R - x)^2 / (G * m) --------------(3)
According to the principle we established earlier, since the force of gravity inside the Earth only depends on the mass above the point x, the mass Mx is equal to the mass of the part of the Earth located above the point x.
To simplify our calculation, we can assume that the Earth's density is constant. In this case, the mass of a sphere located at a distance x from the center of the Earth is given by:
Mx = (4/3) * π * (R^3 - x^3) * ρ --------------(4)
where
ρ is the density of the Earth.
Setting equations (3) and (4) equal to each other, we can solve for x:
(W - 5.5) * (R - x)^2 / (G * m) = (4/3) * π * (R^3 - x^3) * ρ
Simplifying further and solving for x in terms of the given parameters, we can find the approximate distance you would have to drill into the Earth.