The center of a 1.50 diameter spherical pocket of oil is 1.20 beneath the Earth's surface. Estimate by what percentage directly above the pocket of oil would differ from the expected value of for a uniform Earth? Assume the density of oil is 800 kg/m^3. Solve for delta g/ g = %
To estimate the difference in the gravitational acceleration above the pocket of oil compared to a uniform Earth, we need to calculate the change in gravitational acceleration caused by the mass of oil. We can use the formula for the gravitational acceleration due to a point mass:
g = (G * M) / r^2
Where:
- g is the gravitational acceleration
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- M is the mass of the object
- r is the distance between the center of the Earth and the center of the object
To solve for the difference in gravitational acceleration, we'll calculate the gravitational acceleration due to the entire Earth (g_uniform), and then subtract the gravitational acceleration due to the oil pocket (g_oil). Finally, we'll express the difference as a percentage of the expected value (g_uniform).
Step 1: Calculate the mass of the Earth
To get the mass of the Earth, we'll use the formula: M = (4/3) * π * ρ * R^3
Where:
- M is the mass of the Earth
- ρ is the average density of the Earth (approximately 5,515 kg/m^3)
- R is the radius of the Earth (approximately 6,371 km or 6,371,000 meters)
Step 2: Calculate the distance between the center of the Earth and the pocket of oil
The depth of the pocket of oil is given as 1.20 meters. Since the diameter of the spherical pocket is 1.50 meters, the radius (r_oil) of the pocket can be calculated as 0.75 meters (1.5 / 2).
Step 3: Calculate the gravitational acceleration due to the Earth (g_uniform)
Using the formula g = (G * M) / r^2, plug in the values of M (from Step 1) and r (the distance from the center of the Earth to the surface).
Step 4: Calculate the gravitational acceleration due to the oil pocket (g_oil)
Using the formula g = (G * M_oil) / r_oil^2, plug in the values of M_oil (which can be calculated by multiplying the volume of the oil pocket by its density) and r_oil (from Step 2).
Step 5: Calculate the difference in gravitational acceleration (delta_g) as g_uniform - g_oil.
Step 6: Calculate the percentage difference(delta_g_percent):
delta_g_percent = (delta_g / g_uniform) * 100
By following these steps, you should be able to calculate the difference in gravitational acceleration above the pocket of oil relative to a uniform Earth and express it as a percentage.