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course PHY 241
01/13/20118:36 pm
Gold Ball
How can we tell that there’s not a gold ball of diameter 1000 meters, just under the ground below the physics lab?
Look up the density of gold, then figure out how much gravitational force that ball would exert on a 1 kg mass in the lab.
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Assume the 1 kg mass is one meter above the floor of the lab
Density of gold = 19.3 g/cm^3
Volume of sphere 1000 meters in diameter = (4/3)pi r^3
1.67 E11pi cm^3
mass of gold sphere roughly equals: 1.01055E13 grams
force of gravitation = (G m1m2)/r^2
G = 6.67 E-11
M1 = 1 kg
M2 = 1.01055E10 kg
R = 1 meter
Force = (6.67E-11 * 1 kg * 1.01055E10 kg)/1 m^2
6.74037E-1 N
Gravitational force = 0.674037 N
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Good reasoning. However the radius of the ball is 500 meters, so the center of the ball is 500 meters away from the 1 kg mass.
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Assuming that the density of the 1000-meter diameter sphere just below the ground is 2500 kg / m^3 (which is what it would be if the ball is typical earth-crust material), how much force does it exert on that 1-kg mass in the lab?
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1.67E11pi cm^3
Mass of earth-crust sphere roughly equals: 1.31161E7
Force = (6.67E-11 * 1 kg * 1.31161E7 kg)/1 m^2
8.74844E-4 N
Gravitational force = 0.0008748 N
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Again the radius of that ball is 500 m, so the 1 kg mass is 500 m from its center.
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What’s the difference in these forces?
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0.673162 Newtons
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Could we detect the difference in the lab? If so, how?
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A simple experiment to find out would be nearly impossible, but it could possibly be done by dropping a small metal ball several times in the lab, recording the results, and repeating the process outside the lab. If the lab does indeed have such a sphere beneath it, it should take less time (albeit only SLIGHTLY less) for the ball to drop in the lab than to drop outside it.
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That experiment would be capable of detecting a .67 N force, which would increase the acceleration of a 1 kg object by about .68 m/s^2.
However the force is much smaller than that. With sensitive equipment it would be possible to measure the difference in the acceleration of a dropped object, though other alternatives are also possible.
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You'll need to correct the last step of your force calculations, but this shouldn't take you more than a minute or two.
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