A spacecraft of mass m is in a clockwise circular orbit of radius R around Earth.. The mass of Earth is ME. Derive an equation for the orbital period T of the spacecraft in terms of m, ME, R, and physical constants, as appropriate.

An exotic planet Vogsphere is known to have a mass that is 1/81 that of the Earth and a radius 0.25 that of the Earth. Astrophysicist Trillian built a rocket and decided to leave the planet and never to return. Given that the escape speed from the Earth is 11.2 km/s, with what speed must Trillian achieve his goal?

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8 ✕ 1011 solar masses. A star orbiting near the galaxy's periphery is 6.0 ✕ 104 light years from its center. (a) What should the orbital period (in y) of that star be?  y (b) If its period is 6.9 ✕ 107 y instead, what is the mass (in solar masses) of the galaxy? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies.  solar masses

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Two concentric spherical shells with uniformly distributed masses M, and M, are situated as shown in the figure below. a M1 M2 Find the magnitude of the net gravitational force on a particle of mass m, due to the shells, when the particle is located at the following radial distances. (Use any variable or symbol stated above along with the following as necessary: a, b, c, and G for the gravitational constant.) (a) r = a F = (b) r = b F = (c) r = c F =

A 750-kg satellite is in a circular orbit about Earth at a height above Earth equal to Earth's mean radius. (a) Find the satellite's orbital speed. 1768.95 × Your response differs from the correct answer by more than 10%. Double check your calculations. m/s (b) Find the period of its revolution. 12.576 Your response differs from the correct answer by more than 100%. h (c) Find the gravitational force acting on it. 114.11 × Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. N

The planet jupiter is roughly 318 times more massive than the planet earth, and it orbits at an average radius of 5.24 times the orbital radius of the earth. Assuming circular motion of both planets about the sun, find: (a) The ratio of the gravitational forces Fe and Fj on the planets due to the sun. (b) Find the ratio of the orbital periods TE to Tj (c) How long is Jupiter's year, measured in earth years

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8 x 1011 solar masses. A star orbiting near the galaxy's periphery is 5.6 x 104 light years from its center. (For your calculations, assume that the galaxy's mass is concentrated near its center.) (a) What should the orbital period of that star be? yr (b) If its period is 5.1 x 107 years instead, what is the mass of the galaxy? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies. solar masses

11 Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8.0 x 10- solar masses. A star orbiting near the galaxy's periphery is 5.9 x 10" light-years from its center. (a) What should the orbital period (in y) of that star be? y. (b) If its period is 6.1 x 10' years instead, what is the mass (in solar masses) of the galaxy? Such calculations are used to imply the existence of other matter, such as a very massive black hole at the center of the Milky Way. solar masses

The gravitational force exerted by the earth on a unit massat a distance r from the center of the planet is   where M is the mass of the earth,R is its radius, and G isthe gravitational constant. Is F a continuous function of r ?