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Gravitation, centripetal acceleration, and weight

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Planet X rotates in the same manner as the earth, around an azis through its north and south poles, and is perfectly spherical. An astronaut who weighs 943.0 N on the earth weighs 915.0 N at the north pole of Planet X and only 850.0 N at its equator. The distance from the norht pole to the quator is 18,850 km, measured along the surface of Planet X.

1. How long is the day on Planet X?

2. If a 45,000 kg satellite is placed in a cicular orbit 2000 km above the surface of Planet X, what will be its orbital period?

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A detailed solution is given for gravitation, centripetal acceleration, and weight.

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The difference in weight at the equator and the north pole is due to the centripetal acceleration. Denote the mass of the astronaut by m, the planet's radius by R and the angular rotation speed by omega. Then the decrease in weight as the astronaut moves to the equator is m omega^2 R. This yields the equation:

m omega^2 R = (915.0 - 850.0) N = 65.0 N (1)

The mass m is known because you know that on earth the astronaut weighs 943 N:

m * 9.81 m/s^2 = 943.0 N -->

m = 96.13 Kg (2)

Inserting (2) in (1) gives:

omega^2 R = 0.676 m/s^2 (3)

The distance from the north pole to the equator along the surface is a quarter of an entire circle around the planet from the north pole to the south pole and back, so the distance in terms of R is 2 pi ...

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