Final answer:
To calculate the gravitational acceleration gd at a distance of 3300.0 km from the Earth's center, first calculate Earth's mass using its density, and then apply Newton's Law of Universal Gravitation.
Step-by-step explanation:
To find the value of the gravitational acceleration gd at a distance d = 3300.0 km from the Earth's center, we start by calculating the mass of the Earth using the given average density ρ=5540.0 kg/m³ and the fact that acceleration due to gravity at Earth's surface g is approximately 9.81 m/s². The formula to calculate the Earth's mass M is given by the volume multiplied by the density: M = 4/3 π R³ ρ, where R is Earth's radius. After finding the mass, use Newton's Law of Universal Gravitation to calculate gd using the formula gd = GM/d², where G is the gravitational constant (6.674×10⁻¹¹ N·m²/kg²).
The calculations will use the radius of the Earth converted to meters (R = 6,371,000 m), since the density is given in kg/m³. At distance d = 3300 km (or 3300×10³ m), we can then calculate gd. Keep in mind that only the mass within radius d contributes to the gravitational acceleration at that point, according to the Shell Theorem.
Assuming that the Earth's density is uniform, the calculation simplifies as we can treat all of the mass as if it's concentrated at the center, which allows us to use the formula directly without having to consider variations in density with depth.