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course Phy 201
`q001. Using the fact that the gravitational force between two given particles is inversely proportional to the square of the distance between them, and that the gravitational force exerted on an object the Earth is inversely proportional to its distance from the center of the Earth:What would be the ratio of the force on a given object located two Earth radii from the center, to the force exerted on the same object at a distance of one Earth radii from the center?
1/4
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What would be the ratio of the force on a given object located three Earth radii from the center, to the force exerted on the same object at a distance of one Earth radii from the center?
1/9
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What would be the ratio of the force on a given object located two Earth radii from the center, to the force exerted on the same object at a distance of three Earth radii from the center?
2 1/4
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Good.
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Sketch a circle representing the Earth, and sketch points representing a certain object at distances of 1, 2 and 3 Earth radii from the center. For each position sketch a vector representing the force exerted on the object at that position. The relative lengths of your vectors should be approximately in proportion to the forces.
According to your sketch what is the approximate length of each of the other vectors, as percents of the length of the longest?
2/3, 1/3
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If the second and third forces are 1/4 and 1/9 as great as the first, the vectors representing those forces should also be 1/4 and 1/9 as great.
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`q002. A certain object has mass 800 kilograms.
How much force does gravity exert on it at the surface of the Earth?
F=ma
(800kg)(9.8m/s^2)
F = 7,840 N
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How much force does gravity exert on it at a distance of two Earth radii from the center of the Earth?
7,840 N*1/4 = 1,960N
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How much force does gravity exert on it at a distance of three Earth radii from the center of the Earth?
1/9*7,840N = 871.1N
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`q003. Reasoning in terms of the inverse square proportionality, how much force would the Earth exert on the 800 kg object of the previous problem if it was located 1.4 Earth radii from the center?
1/1.96*7840N = 4000N
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Assume Earth to have radius 6400 km (this is in fact about half a percent greater than Earth's average radius).
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How far from the center of the Earth would an object be if it was at 1.4 Earth radii from the center?
6400km*1.4radii = 8960km
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Using F = G M m / r^2, where G = 6.67 * 10^-11 N m^2 / kg^2, calculate the force exerted on the object at this position. The mass of the Earth is about 6 * 10^24 kg.
F = (6.67 * 10^-11 N m^2 / kg^2)(6 * 10^24 kg)(800kg)/(8960km)^2
F = 3,987,962,372N
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F = 3,987,962,372 N * m^2 / km^2, not 3,987,962,372N.
m^2 / km^2 = m^2 / (1000 m)^2 = m^2 / (1 000 000 m^2) = 1/1 000 000.
This would give you a force of 3987.9 N, which is equal up to roundoff error to the 4000 N you reasoned out previously.
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