Careful measurements of local variations in the acceleration due to gravity can reveal the locations of oil deposits. Assume that the Earth is a uniform sphere of radius 6370 km and density 5500 kg/m3, except that there is a spherical region of radius 1.7 km and density 944 kg/m3, whose center is at a depth of 3.4 km. Suppose you are standing on the surface of the Earth directly above the anomaly with an instrument capable of measuring the acceleration due to gravity with great precision. What is the ratio of the acceleration due to gravity that you measure compared to what you would have measured had the density been 5500 kg/m3 everywhere? (Hint: Think of this as a superposition problem involving two uniform spherical masses, one with a negative density.)
To determine the ratio of the acceleration due to gravity at the surface of the Earth above the oil deposit anomaly compared to what it would have been without the anomaly, we need to consider the gravitational pull exerted by both the uniform Earth and the spherical region with negative density.
Here's how you can calculate it:
1. First, let's determine the gravitational acceleration due to the uniform Earth of density 5500 kg/m3. The mass of the Earth, M, can be calculated using the formula:
M = (4/3) * π * r^3 * ρ,
where r is the radius of the Earth and ρ is the density.
Substituting the values, we have:
M = (4/3) * π * (6370 km)^3 * 5500 kg/m3.
2. Next, we need to calculate the gravitational acceleration on the surface due to the uniform Earth using Newton's law of gravitation:
g_Earth = G * M / r^2,
where G is the gravitational constant and r is the radius of the Earth.
Substituting the values, we have:
g_Earth = G * M / (6370 km)^2.
3. Now, let's calculate the gravitational acceleration due to the spherical region with negative density. We can think of this as a negative mass since it has a lower density compared to the Earth's density.
The mass of the region, M_anomaly, is given by:
M_anomaly = (4/3) * π * (1.7 km)^3 * (-944 kg/m3).
4. The gravitational acceleration due to the spherical anomaly at the surface can be calculated using Newton's law of gravitation as well:
g_anomaly = G * M_anomaly / (r + 3.4 km)^2,
where r is the radius of the Earth.
5. Finally, we can obtain the ratio of the measured acceleration due to gravity to what it would have been without the anomaly by adding the individual gravitational accelerations:
g_ratio = (g_Earth + g_anomaly) / g_Earth.
Substitute the values into the equations and calculate the final result.