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 = %
do it this way.
you know g for a solid earth.
Now find g for a solid sphere of Earth the size of the bubble, 1.20km away.
Then, find g for a sphere of oil the size of the bubble, 1.20km away.
you have three numbers:
Take the first, subtract the second, and add the third. That is the resultant g value.
Remember in working these fictional bubbles, the mass is volume*density.
g'=GM'/distance^2 where M' is the mass of the bubble (Earth, then oil).
To estimate the percentage by which the gravitational field strength above the pocket of oil differs from the expected value for a uniform Earth, we can use the concept of the gravitational field strength due to a point mass.
The equation for gravitational field strength is given by:
g = (G * M) / r^2,
where:
- g is the gravitational field strength,
- G is the universal gravitational constant (approximately 6.67 x 10^-11 Nm^2/kg^2),
- M is the mass of the object creating the gravitational field, and
- r is the distance between the object and the point where the field is being measured.
In this case, the pocket of oil can be treated as a point mass.
1. First, we need to calculate the mass of the pocket of oil.
The volume of the spherical pocket of oil can be determined using the formula:
V = (4/3) * π * r^3,
where r is the radius of the pocket (which is half the diameter). So, in this case, r = 1.50/2 = 0.75 m.
Given that the density of oil is 800 kg/m^3, we can calculate the mass using the formula:
m = density * V.
Plugging in the values, we have:
m = 800 kg/m^3 * [(4/3) * π * (0.75 m)^3] = 1413.716 kg.
2. Next, we need to calculate the gravitational field strength at the depth of the pocket of oil.
Using the equation g = (G * M) / r^2, we find:
g = (6.67 x 10^-11 Nm^2/kg^2 * 1413.716 kg) / (1.20 m)^2 = 4.139 m/s^2.
3. Finally, we need to find the expected gravitational field strength on the surface of a uniform Earth.
Assuming a uniform Earth, we can use the formula g = G * M_earth / r_earth^2,
where M_earth is the mass of the Earth (approximately 5.972 x 10^24 kg),
and r_earth is the radius of the Earth (approximately 6.371 x 10^6 m).
Plugging in the values, we have:
g = (6.67 x 10^-11 Nm^2/kg^2 * 5.972 x 10^24 kg) / (6.371 x 10^6 m)^2 = 9.819 m/s^2.
4. Finally, we can calculate the percentage difference in the gravitational field strength from the expected value.
The percentage difference is given by:
(delta g / g) * 100.
Plugging in our calculated values:
(delta g / g) = (9.819 m/s^2 - 4.139 m/s^2) / 9.819 m/s^2 = 0.578,
so the percentage difference is:
(delta g / g) * 100 = 0.578 * 100 = 57.8%.
Therefore, we can estimate that the gravitational field strength directly above the pocket of oil would differ from the expected value for a uniform Earth by approximately 57.8%.