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.
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 the universal law of 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 × 10^-11 N m^2/kg^2),
m1 and m2 are the masses of the two objects, and
r is the distance between the centers of the two objects.
In this case, the gravitational force exerted on the astronaut is equal to their weight (90 N), and the mass of the Earth is given (6 x 10^24 kg). The mass of the astronaut is also given (90 kg).
Let's plug in the values in the equation and solve for r:
90 N = (G * 90 kg * 6 x 10^24 kg) / r^2
Simplifying the equation:
90 N * r^2 = (G * 90 kg * 6 x 10^24 kg)
r^2 = (G * 90 kg * 6 x 10^24 kg) / 90 N
r^2 = G * 6 x 10^24 kg
Now, divide both sides of the equation by G:
r^2 = (6 x 10^24 kg * G) / 90 kg
r^2 = (6 x 10^24 kg * 6.67430 × 10^-11 N m^2/kg^2) / 90 kg
Simplify the equation by canceling out units:
r^2 = 4.4482 × 10^12 m^3/kg
Now, take the square root of both sides to find r:
r = sqrt(4.4482 × 10^12 m^3/kg)
Calculating the square root (approximately):
r ≈ 2.363 × 10^6 meters
Therefore, the astronaut must be approximately 2.363 × 10^6 meters away from the center of the Earth in order for them to weigh 90 N.