Three uniform spheres of masses m1 = 2.50 kg, m2 = 4.00 kg, and m3 = 6.50 kg are placed at the corners of a right triangle (see figure below). Calculate the magnitude of the resultant gravitational force (in 10-11 N) on the object of mass m2, assuming the spheres are isolated from the rest of the Universe.

Please answer the equation step by step and include a proper diagram if needed. Usse gravity as 9.8m/s^2. 4. Mars and Earth, at their closest theoretical positions, are 54.6 million km apart. The mass of Earth is 5.97 x 10 kg, and the mass of Mars is 6.39 x 10ª kg. A satellite of mass 9550 kg is to be placed such that the forces of gravity between the satellite and the two planets are equal in magnitude. Determine two possible locations for this to occur.

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 5.9 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 5.2 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

A particle of mass m is placed at a distance r away from the center of a thin circular hoop of mass M and radius R. The particle is in the plane of the hoop, and r < R. M. Find the gravitational force on the particle. Does your answer make sense in the limit r + 0? r + R? 2.