Question 6) Oil and gas exploration companies find new underground deposits by measuring (with incredible precision) the value of g at the Earth's surface.
EXTRA INFO GIVEN:
The challenge: F=Gm1m2/r^2 ONLY works for two cases: point masses, or if one of the masses has spherical symmetry (remember our discussions in class about the force of gravity due to a hollow shell.) For the case of a mass with spherical symmetry, the "r" refers to the distance bewteen the CENTRE of the two masses. That is why when you calculate the gravity on a space ship 300 km above the surface of the Earth, you use r=Re+300 km (where Re is the radius of the Earth).
The mass in the oil pocket problem does NOT have spherical symmetry but can you break it into smaller problems, each of which does have spherical symmetry.
E.g: 1) can you find the force of gravity if the Earth was a solid sphere with uniform density? Look up the radius of the Earth and its mass.
2) can you find the force of gravity from rock (with the density of the Earth) shaped like a sphere with radius 0.5 km that centre is 1 km away from you?
3) can you find the force of gravity from a sphere of oil with radius 0.5 km that centre is 1 km away from you?
4) considering these three results, how do you now answer the question asked? [Hint: do NOT just add all three forces together!.
One other point, in class all of you had no problem determining the density of the Earth (total mass/total volume). In the drop-in centre, some students did not realize that this allowed them to find the mass of any shape of rock (with volume V): M =density * V, where density can be found from total Mass/total Volume
To find new underground deposits using the value of g at the Earth's surface, oil and gas exploration companies use the principle of gravity. The force of gravity between two objects is given by the equation F = G * m1 * m2 / r^2, where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers.
However, this equation only works for two cases: point masses or if one of the masses has spherical symmetry. In the case of the Earth, the mass can be considered to be concentrated at its center, so the distance "r" refers to the distance between the centers of the Earth and the other object.
When dealing with underground deposits, the mass distribution is not spherically symmetric. However, we can break down the problem into smaller problems, each of which does have spherical symmetry. Here's how we can approach it:
1) Find the force of gravity if the Earth was a solid sphere with uniform density: You can look up the radius of the Earth (Re) and its mass to calculate this force. Remember that the distance "r" is the distance between the center of the Earth and the object you are measuring the force on.
2) Find the force of gravity from rock (with the density of the Earth) shaped like a sphere: Consider a rock with a radius of 0.5 km and a center located 1 km away from you. Use the same formula to calculate the force of gravity between you and the rock, assuming the rock has the same density as the Earth.
3) Find the force of gravity from a sphere of oil: Consider a sphere of oil with a radius of 0.5 km and a center located 1 km away from you. Again, use the same formula to calculate the force of gravity between you and the oil.
4) Considering the results from steps 1, 2, and 3, you need to determine how to answer the question asked. Hint: Do not just add all three forces together. You need to take into account the distribution of mass and the different distances involved.
In class, you should have no problem determining the density of the Earth (total mass/total volume). This allows you to find the mass of any shape of rock using the equation M = density * V, where density can be found from the total mass/total volume.
By following these steps and considering the different forces of gravity calculated, you can determine the presence of underground deposits based on the measured value of g at the Earth's surface.