Intro Geophysics
Gravity Homework Problems

Assigned:7 February
Due:13 February

  1. Suppose that somewhere in space the gravitational potential is known to follow the function U(x,y,z) = 3x4 + 3y4 + 4cos(z). What is the gravitational acceleration of due to this potential field at point (x = 1, y = 2, z = pi/2 = 3.14159/2.)?

  2. Christopher often goes to Boston from RPI for the weekend and is curious to know if he gains any weight on these trips. Being a very dedicated and hard working student, he doesn't have time to weigh himself before and after the trip and so takes some scales with him and weighs himself on the bus ride on the way home and on the way back. In Geophysics class on Monday he mentions to Sam that he weighed exactly the same amount on both trips, and so concludes that his weight didn't change. "Au contraire", says Sam, "if you weighed yourself before and after the trip in the comfort of your own bathroom here at RPI, you'd see a difference". What would Christopher observe if he took Sam's advice, and why?

  3. The EXACTO Engineering company advertizes that they can build a perfectly straight building by using lasers to ensure that their steel posts are exactly aligned and perpendicular to the beams. You contract them to build a 100 story skyscraper in downtown Albany. When it's finished, the tenants on the top floor complain that their floor is not level and that various objects are rolling off their desks. You take EXACTO to court to sue them for poor construction, but EXACTO argues that they did exactly what they said they would do. They also point out that the tenants on the ground floor are not having any problems at all. The judge not only agrees with EXACTO, but also says that the upper floor tenants are right and they should be compensated by you. Explain how you were so unfortunate as to loose both sides of this case.

  4. Showing true entrepreneurial spirit, Jackie capitalizes on her Geophysical knowledge to open a business that combines a travel agency with a weight loss program. Her company, "Bound to Lose", offers a flight from Albany to Jamaica that they claim results in a guaranteed instant loss of weight (or triple your money back, they offer). How can she be so sure, and how much weight would a 250 lb person lose just by taking this trip?

  5. Katherine wants to measure the density of a rock in her field area. "It's easy", she says, "all you need to do is measure the mass of the rock first in air and then submerged in water. And you don't need to know anything about the change in the level of the water - you really only need to know the difference in the measured mass!". Show that Katherine knows what she's talking about.

  6. Lisa is planning a gravity survey of the RPI campus, and wants to be sure her readings are accurate to 0.1 mgal. To what accuracy does she need to know the location and elevation at the points where she measures gravity?

  7. In addition to elevation, for the survey mentioned in the previous problem Lisa needs to have some idea of density of the rocks near the surface in order to make a correct Simple Bouguer Correction. If she is 10 meters above her reference level, and wants to correct her readings to this level, how well must she know this density in order for his data to be accurate to 0.1 mgal?

  8. The following gravity data were collected near by a Geophysics class near Glens Falls, NY. Using the information in the table, calculate the Simple Bouguer Anomaly at stations S02 and S03 relative to the base. Ignore tidal corrections, and assume a density of 2.0 gr/cm3. Note: The units of UTM North and East are in kilometers.

    StationUTM NorthUTM EastElevation
    (ft)    		Reading
    (mgals)
    BASE4792.03858603.83929379.29153.486
    S024792.05811603.86188379.35153.251
    S034792.07764603.88538379.47152.955

  9. Taliban insurgents attempt to travel from Pakistan to Afghanistan by digging a cylindrical tunnel 2 meters in diameter under the border. The top of the tunnel is 1 meter below the surface. Because he has a reputation as an excellent Geophysicist, Lance is contracted by the US military to find this tunnel using a gravity meter. How accurate must his readings be in order to detect the tunnel (in the sense that you will need to be at least as accurate as the maximum value of the signal produced by the tunnel)? Assume that the rocks through which the tunnel was dug have a uniform density of 2.67 gr/cm3. (Hint: Use the formula for the gravity signal due to a horizontal cylinder given in the text).

  10. Angela is reducing gravity data she collected on a field trip to Nebraska. Normally this would be a breeze (as Nebraska is very flat) but there are several sites located right at the edge of a large, 10 meter deep swimming pool. In fact, the pool is so large that for present purposes you can think of it as a semi-infinite sheet. What terrain correction should Angela apply to her data if the pool is empty, and what should it be if the pool is full of water? Assume the density of the ground around the pool is 2 gr/cm3, and that the density of the water is 1 gr/cm3.