How far from the center of the Earth must a 90 kg astronaut be in order for them to weigh 90 N? The mass of the Earth is 6 x 1024 kg.

In order to find the distance from the center of the Earth where the astronaut would weigh 90 N, we can use the concept of gravitational force and the equation for Newton's law of universal gravitation.

The equation for the gravitational force between two objects is given by:

F = (G * m1 * m2) / r^2

Where F is the gravitational force, G is the gravitational constant (approximately 6.67430 x 10^-11 N m^2/kg^2), m1 and m2 are the masses of the two objects in kilograms, and r is the distance between the centers of the two objects in meters.

In this case, we know that the gravitational force (F) acting on the astronaut is 90 N, the mass of the astronaut (m2) is 90 kg, and the mass of the Earth (m1) is 6 x 10^24 kg. We need to find the distance (r) from the center of the Earth where this force is acting.

Let's rearrange the formula to solve for r:

r^2 = (G * m1 * m2) / F

Substituting the given values:

r^2 = (6.67430 x 10^-11 N m^2/kg^2 * 90 kg * 6 x 10^24 kg) / 90 N

Calculating this equation gives us:

r^2 = 4.44980 x 10^13 m^2

To solve for r, we take the square root of both sides:

r = √(4.44980 x 10^13 m^2)

Performing the calculation gives us:

r ≈ 6.673 x 10^6 meters

Therefore, the distance from the center of the Earth where the astronaut would weigh 90 N is approximately 6.673 x 10^6 meters.