Final answer:
The given relation p=3g/(4πRG) for Earth's density is not dimensionally consistent because the dimensions of the right side do not match those of density. The gravitational constant G, however, is dimensionally consistent within gravitational calculations.
Step-by-step explanation:
To check the dimensional consistency of the given relation p=3g/(4πRG), we need to compare the dimensions of each term on both sides of the equation. The density p on the left side has dimensions of mass per unit volume (M/L^3), while the right side contains the gravitational constant G with dimensions of (M^-1L^3T^-2), the radius of Earth R with dimensions of length (L), and g, which typically refers to acceleration due to gravity and has dimensions of (LT^-2). The dimensions on the right side thus become M/(L^3 * M^-1L^3T^-2 * L * LT^-2) = M/(L^2 * T^-4), which are not equal to the dimensions of density (M/L^3). Therefore, the relation is not dimensionally consistent and is not correct.
Using the known value of the gravitational constant G (6.67 × 10^-11 N·m²/kg²), the mass of the Earth (5.971×10^24 kg), and the radius of Earth (6.371x10^6 m), we see that the dimensions of G match those required for consistency in calculations related to gravity, demonstrating that G is dimensionally consistent.