A rock on Mars has a mass of 1200 kg. At midday, the rock is directly between the sun and Mars. i) Determine the force exerted on the rock by the sun. Take the distance from the rock to the sun's center as 2.28 × 10¹¹ m. ii) Contrast this force with the force on the rock from Mars (g_M = 3.72 m/s²).

Four celestial objects, each possessing a mass equivalent to the Earth, arrange themselves into a square configuration, with each side measuring 1.4 x 10¹¹ m. In order to maintain this structure and prevent a gravitational collapse toward the center, the square must uniformly rotate. What is the duration of one complete rotation period?

A cylindrical object with a mass of 2.3 kg is located on a satellite that orbits Mars at an altitude of 400 km from its surface. Determine the gravitational force experienced by this cylinder. The mass of Mars is (6.42 × 10²³ kg) and its radius is  (3.34  × 10⁶ m).

Consider a system of clay balls of different masses placed at various positions in space, as shown in the figure. In the given scenario, calculate the net gravitational force on the 25 kg mass. Express your answer in unit vectors.

What is the percentage decrease in gravity strength at an altitude of 400 km above the Earth's surface? Given values are R_Earth=6.38×10⁶ m, G=6.67×10^-11 N.m²/kg², M_Earth=5.97×10²⁴ kg.

Archeologists place a pendulum on the surface of the Earth at a distance of D away from the center of a gigantic pyramid of mass M_p. The bob of the pendulum (having mass m) makes an angle of Φ with the vertical at the equilibrium due to the presence of the pyramid. Given that the radius of the Earth is R_E and the mass of the Earth is M_E, evaluate an expression of Φ in terms of M_p, D, R_E, and M_E.

Consider a celestial system where two stars, each of identical mass, orbit around their common center of mass which is located precisely at the midpoint of the line joining them. These stars are separated by 10 × 10¹¹ m. They complete one full orbit around their common center of mass every 12.6 years. Determine the mass of each star in this dual-star system.