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

Uniform Density Sphere

This concept involves assuming that the Earth (or any planet) has a constant density, which greatly simplifies the calculation of gravitational forces both on the surface and within the interior. Under this model, the mass enclosed within any radius is proportional to the cube of that radius, leading to key results such as the linear variation of gravitational acceleration with distance from the center.

Gravitational Acceleration Within a Uniform Sphere

Thanks to the shell theorem, only the mass enclosed by a sphere of radius equal to the distance from the center contributes to the gravitational field at that point. In a uniform density sphere, this means that the gravitational acceleration inside is directly proportional to the radial distance from the center, reaching its maximum at the surface and zero at the center.

Gravitational Acceleration Reduction Factor

This concept involves comparing the gravitational acceleration at two different points by using a ratio. By knowing that gravitational acceleration inside a uniform sphere varies linearly with radius, one can determine how deep one must go so that the acceleration is a specified fraction (such as one half) of its surface value.

Percentage Difference in Gravitational Acceleration

This refers to calculating the relative change in gravitational acceleration between two positions, typically expressed as a percentage. It is found by taking the difference of the two accelerations, dividing by the reference acceleration (often the one at the surface), and converting the result into a percentage. This measure helps to quantify how significant the change in gravity is due to depth.

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