Final answer:
The universal constant of gravitation (G) in terms of the time period (T) of a satellite's orbit and the average density (ρ) of the planet is represented by the expression 4π²/T²ρ, according to the law of universal gravitation and Kepler's third law.
Step-by-step explanation:
The question is asking for the expression of the universal constant of gravitation (G) in terms of the time period (T) of a satellite orbiting a planet and the average density (ρ) of the planet.
From the information provided and the law of universal gravitation, we know that for orbiting bodies, Kepler's third law applies, which states that the ratio of the cube of the orbit radius (r³) to the square of the period (T²) is a constant. This ratio is given by r³/T² = GM/4π², where G is the universal constant of gravitation and M is the mass of the orbited body.
If we express M in terms of the density ρ and the volume of the spherical planet (assuming it's a sphere), M = ρ * (4/3)πr³. So, we get r³/T² = (4π²/G)(ρ * (4/3)πr³). Simplifying further, we find that G = 4π²/T²ρ. Therefore, the correct option that represents the universal constant of gravitation G in terms of T and ρ is C. 4π²/T²ρ.