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Sensitive meters that measure the local free-fall acceleration g can be used to detect the presence of deposits of near-surface rocks of density significantly greater or less than that of the surroundings. Cavities such as caverns and abandoned mine shafts can also be located. (a) Show that the vertical component of g a distance x from a point directly above the center of a spherical cavern (see Fig) is less than what would be expected, assuming a uniform distribution of rock of density p, by the amount where R is the radius of the cavern and d is the depth of its center. (b) These values of ∆g, called anomalies, are usually very small and expressed in milligals, where 1 gal = 1 cm/s². Oil prospectors doing a gravity survey find ∆g varying from 10.0 milligals to a maximum of 14.0 milligals over a 150-m distance. Assuming that the larger anomaly was recorded directly over the center of a spherical cavern known to be in the region, find its radius and the depth to the roof of the cavern at that point. Nearby rocks have a density of 2.80 g/cm 3 , (c) Suppose that the cavern, instead of being empty, is completely flooded with water. What do the gravity readings in (b) now indicate for its radius and depth?

Sensitive meters that measure the local free-fall acceleration g can be used to detect the presence of deposits of near-surface rocks of density significantly greater or less than that of the surroundings. Cavities such as caverns and abandoned mine shafts can also be located. (a) Show that the vertical component of g a distance x from a point directly above the center of a spherical cavern (see Fig) is less than what would be expected, assuming a uniform distribution of rock of density p, by the amount
where R is the radius of the cavern and d is the depth of its center. (b) These values of ∆g, called anomalies, are usually very small and expressed in milligals, where 1 gal = 1 cm/s². Oil prospectors doing a gravity survey find ∆g varying from 10.0 milligals to a  maximum of 14.0 milligals over a 150-m distance. Assuming that the larger anomaly was recorded directly over the center of a spherical cavern known to be in the region, find its radius and the depth to the roof of the cavern at that point. Nearby rocks have a density of 2.80 g/cm3, (c) Suppose that the cavern, instead of being empty, is completely flooded with water. What do the gravity readings in (b) now indicate for its radius and depth?

Grade:11

1 Answers

Kevin Nash
askIITians Faculty 332 Points
6 years ago
One can find\deltag by pretending the Earth is not there, but the presence material in the hole.
Concentrate on the vertical component of the resulting force of attraction. Then,
234-638_f14.JPG
Here, the straight line distance from the prospector to the center of the hole is r and the mass of the material that would fill the hole is M.

As the density of the rock is uniform, the mass of the hole is
234-61_f15.JPG
Here, density of the rock is .
Solution:
(a)
The given figure can be represented as shown below:

234-1782_f16.JPG
234-298_f17.JPG
(b)
234-1609_f18.JPG
234-2128_f18.JPG

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