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Chapter 12: Problem 35

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 \(\left(a_{\text {surf }}-a_{3,5 \mathrm{~km}}\right) / a_{\text {surf }} ?\)

Short Answer

Expert verified
Answer: To find the depth at which the gravitational acceleration is reduced by half, first calculate the surface acceleration using the formula \(g_{surf} = -\frac{4}{3} \pi G \rho R\). Then, set up an equation to solve for the depth \(D\) such that \(g(R - D) = \frac{1}{2} g_{surf}\). Calculate the value of \(D\). To find the percentage difference in acceleration between the surface and a 3.5 km deep mine shaft, first calculate the gravitational acceleration at 3.5 km depth using \(g_{3.5 \text{ km}} = -\frac{4}{3} \pi G \rho (R - 3.5)\). Then, compute the percentage difference using the formula \(\frac{g_{surf} - g_{3.5 \text{ km}}}{g_{surf}}\). Express the result as a percentage.

Step by step solution

01

Gravitational Acceleration Formula

To calculate the gravitational acceleration inside a uniform sphere, we can use the formula: \(g(r) = - \frac{4}{3} \pi G \rho r\), where \(g(r)\) is the gravitational acceleration at radius \(r\), \(G\) is the gravitational constant, and \(\rho\) is the density of the Earth.
02

Find the Gravitational Acceleration at the Surface

Calculate the surface acceleration \(g_{surf}\) by using the radius of the Earth, \(R = 6370 \text{ km}\). We will also need the Earth's density \(\rho\). We can estimate \(\rho = 5500 \text{ kg/m}^3\). So, \(g_{surf} = -\frac{4}{3} \pi G \rho R\)
03

Determine the Depth for Half Gravitational Acceleration

We want to find the depth \(D\) such that the gravitational acceleration is half its surface value, i.e., \(g(R - D) = \frac{1}{2} g_{surf}\). To solve for \(D\), set up the equation as follows: \(\frac{1}{2} g_{surf} = -\frac{4}{3} \pi G \rho (R - D)\) Solve for \(D\) and calculate its value.
04

Calculate the Gravitational Acceleration at 3.5 km Depth

Next, we need to find the gravitational acceleration at the bottom of the 3.5 km deep mine shaft. Calculate \(g_{3.5 \text{ km}}\) using the formula: \(g_{3.5 \text{ km}} = -\frac{4}{3} \pi G \rho (R - 3.5)\)
05

Calculate the Percentage Difference in Gravitational Acceleration

Finally, we can calculate the percentage difference between the surface and 3.5 km depth accelerations as follows: \(\frac{a_{\text {surf }}-a_{3,5 \mathrm{~km}}}{a_{\text {surf }}} = \frac{g_{surf} - g_{3.5 \text{ km}}}{g_{surf}}\) Compute the value and express it as a percentage.

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