Some of the deepest mines in the world are in South Africa...

Some of the deepest mines in the world are in South Africa and are roughly $3.5 \mathrm{~km}$ deep. Consider the Earth to be a uniform sphere of radius $6370 \mathrm{~km}$. a) How deep would a mine shaft have to be for the gravitational acceleration at the bottom to be reduced by a factor of 2 from its value on the Earth's surface? b) What is the percentage difference in the gravitational acceleration at the bottom of the $3.5-\mathrm{km}$ -deep shaft relative to that at the Earth's mean radius? That is, what is the value of $\left(a_{\text {surf }}-a_{3,5 \mathrm{~km}}\right) / a_{\text {surf }} ?$

Some of the deepest mines in the world are in South Africa and are roughly $3.5 \mathrm{~km}$ deep. Consider the Earth to be a uniform sphere of radius $6370 \mathrm{~km}$.
a) How deep would a mine shaft have to be for the gravitational acceleration at the bottom to be reduced by a factor of 2 from its value on the Earth's surface?
b) What is the percentage difference in the gravitational acceleration at the bottom of the $3.5-\mathrm{km}$ -deep shaft relative to that at the Earth's mean radius? That is, what is the value of $\left(a_{\text {surf }}-a_{3,5 \mathrm{~km}}\right) / a_{\text {surf }} ?$

Key Concepts

Percentage Difference Calculation

This concept involves comparing the difference between two values relative to a reference value, expressed as a percentage. In the context of gravitational acceleration, it is used to quantify how much lower the gravitational acceleration is at a given depth compared to the acceleration at the surface. This measure provides an intuitive expression of the relative change in gravitational force.

Gravitational Acceleration Inside a Uniform Sphere

In a uniformly dense sphere, the gravitational acceleration at a distance from the center is directly proportional to that distance. This stems from the fact that only the enclosed mass (which increases with the cube of the radius) contributes to the gravitational pull, while the dependence on the square of the distance in Newton’s law results in a linear relationship with the radial distance. This linear variation is critical for understanding how gravity changes with depth.

Uniform Density Model

This concept assumes the Earth (or any massive sphere) has a constant density throughout its volume. By making this simplifying assumption, the gravitational effects inside the Earth become analytically tractable. It allows one to derive a simple linear relation in the interior of the sphere, rather than more complex variations that occur in a body with varying density.

Shell Theorem

The shell theorem states that a spherically symmetric shell of mass exerts no net gravitational force on objects located inside it. This means that when calculating the gravitational acceleration at a point inside a uniform sphere, only the mass enclosed within the radius to that point contributes to the gravitational force, while all the mass outside that radius cancels out.

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