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Gravitation and Orbital Motion (College)

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Gravitation and Orbital Motion

Astronomers using Hubble Space Telescope discover a planet orbiting a distant star. They measure the planet's period of revolution to be equal to 135 days and the average distance between the planet and the star to be equal to [math]2.35xx10^8km[/math]. Calculate the planet's mass using this data.

Two identical asteroids of mass [math]M[/math] are viewed as spheres of radius [math]r[/math]. They are initially at rest at a distance orders of magnitude greater than their radii, and are being pulled towards each other by their own gravity. Assuming that no other objects are acting on them, what is the speed of the asteroids immediately prior to their collision? Assume this speed is much less than the speed of light.

In order to get to the other side of the globe, an explorer digs a straight tunnel that goes through the Earth's core and comes out on the other side. Describe the motion of an object dropped into this tunnel. Neglect air drag. If the object exhibits cyclical motion, calculate its period. Assume the Earth's density to be homogeneous.

One of the theories explaining the formation of Saturn rings shown below states that if an orbiting object gets close to a massive planet, the tidal force exerted on the object's surface will exceed the force of the object's own gravity, leading to the object's disintegration and eventual scattering along its orbit. The maximum distance from the planet for which this is possible is known as Roche limit after French astronomer Edouard Roche who first calculated this limit in 1848. Calculate Roche limit in a simplified case by equating the force of gravity exerted on a probe mass by a planet of mass [math]M_p[/math] and radius [math]R_p[/math] to the net force exerted on the same probe mass by a satellite of mass [math]M_s[/math] and radius [math]R_s[/math] orbiting the planet on a circular orbit of radius [math]d[/math] (Roche limit). Assume that the satellite is orbiting with synchronous rotation, meaning that it makes one rotation about its axis in about the same time it takes it to orbit the planet, much like the Moon orbiting the Earth. Take the centrifugal force of this rotation into account.
Planet Saturn - Small

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