Final answer:
The acceleration due to gravity at a height of 6.4 × 10^6 meters above the Earth's surface is calculated using the formula g = G × (M / r^2), incorporating the gravitational constant G, the mass of the Earth M, and the distance from the Earth's center r.
Step-by-step explanation:
To compute the acceleration due to gravity (g) at a height of 6.4 × 106 meters above the Earth's surface, we need to use the formula for gravitational force:
g = G × (M / r2)
where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth. Given that G = 6.67 × 10-¹¹ N · m2/kg2, M is 5.98 × 1024 kg, and the Earth's radius re is 6.38 × 106 meters, we calculate the new distance r as the sum of the Earth's radius and the given height above the surface:
r = re + height
r = 6.38 × 106 m + 6.4 × 106 m
Computation yields:
r = 1.278 × 107 m
Now, substituting these values into the formula:
g = (6.67 × 10-¹¹ N · m2/kg2) × (5.98 × 1024 kg) / (1.278 × 107 m)2
By calculating the above expression, we can find the value of g at the given height.