GRAVITY and THE SAN FRANCISCO BAY AREA


Exercises"

(A) Drop a ball or use a pendulum to calculate g. For the first part of this exercise, you will need a stopwatch, a meter-stick, and a ball. How long does it take the ball to drop a vertical distance of 3m? Drop the ball and time how long it takes for the ball to hit the ground. Do this 5 times. What kind of range in times did you measure? Why? The relationship between distance, (d), time (t), initial velocity (vo), and gravitational acceleration (g) is the following:

d = vot + 1/2 gt2
Since vot =0,
d=1/2gt2


Using this formula, what value of g did you measure? What are the units of g?

For the second part of this exercise, you will need a stopwatch, a string, a meter-stick, and a ball. Tie the ball to one end of the string. Hang the string so that the ball can swing freely. Measure how long the string is. Holding the string taut, drop the ball so that it swings back and forth like a pendulum. Measure how much time it takes for the ball to return to its initial position. That amount of time is called a period. Improve your precision by measuring the time for several consecutive swings and then dividing the time by the number of swings to determine the period. The relationship between period (T), the length of the string (l), and g is the following:

T=2 pi * (square root(l/g))

You have measured T and l. What values of g do you measure and how do they compare to the values derived in the first exercise? Describe the sources of error in measuring g this way. Note that neither of these equations depends on the mass of the ball.

(B) Measuring densities of rocks (sandstone vs. gabbro). For this exercise, you'll need a balance or scale, a sink full of water, and a wire basket. Geophysicists measure densities by using Archimedes' principle. Instead of weighing the rock and then dividing the weight by the volume that you laboriously measured of an irregularly-shaped rock to get the density, you will weigh the rock in air and then in water. The weight in water reflects the amount of water displaced by the rock's volume. To weigh the rock in water, first weigh the wire basket by suspending it from the balance so that the basket is completely submerged. Now weigh the rock in the basket. Subtract the weight of the rock in the basket from the weight of the basket to get the weight of the rock in water. To get the density, you divide the weight of the rock in air by the weight in air minus the weight in water, wa/wa-ww. This method works because the density of water is exactly 1 gram/cubic centimeter. Measure a sedimentary rock, an igneous rock, and a metamorphic rock. How do the densities vary from rock to rock? Why? How do these densities compare with the values listed in Table 1 (make sure you are using the same units!)?

(C) Write out in mathematical form the relationship between gravitational force, mass, and distance. Newton also described force in a general way: F=ma, where F is the force in question, m is the mass being subjected to the force, and a is acceleration. You can also write that the gravitational force acting on your mass is F=mg. Using the two equations for force, calculate the mass of the Earth, if the Earth's radius, r, is 6370 km, average g is 9.81 m/s2, and g is 6.67 x 10-11 m3/s2kg. Now calculate the average density of the Earth, given that the volume of a sphere is equal to 4/3pr3. How does this average density compare with the densities listed in Table 1?

(D) Assuming that the underlying rock types are the same, would you measure more gravitational acceleration on top of Mt. Diablo or at Fisherman's Wharf at the San Francisco waterfront? Why? What are the isostatic gravity values at these two places? Explain why a larger value of g does not always correspond to a larger value of residual gravity.

(E) The shape of the Earth is not a perfect sphere. The Equator is farther from the center of the Earth than are the poles. Would you weigh more at the Equator or at the North Pole? Would you weigh more on an airplane (in level flight) or in space? Why?

(F) Assuming everything else is equal, would a ball drop faster if you were standing on a mountain made of gold or on a mountain made of

(G) Groundwater is an important source of water for agriculture and drinking water. Water under the ground resides in the spaces (pores) between the grains that make up the rocks as well as in cracks. The more pores a rock has, the less dense it will be. Would you expect to find high or low gravity values over very porous rocks? Looking at GP-1006, where would be good places to find groundwater?

(H) If closely spaced gravity contours indicate a fault that has juxtaposed rocks of differing densities, where would you extend the Hayward fault north across San Pablo Bay?


Answers to Exercises

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