You have probably seen films of astronauts floating weightless in orbiting satellites. People often get the idea that the astronauts are weightless because they are so far from the gravity of the earth. Let us see if that explanation is correct.
Typically, such satellites orbit around 400 above the surface of the earth. If an astronaut weighs 630 on the ground, what will he weigh if he is 400 above the surface?
In light of your answer to part A, are the astronauts weightless because gravity is so weak? Why are they weightless?
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yikers
To determine the weight of the astronaut when they are 400 miles above the surface of the earth, we need to understand how weight changes with distance from the center of the earth.
It is common to assume a constant acceleration due to gravity on the surface of the earth, which is approximately 9.8 meters per second squared (m/s^2). However, using Newton's law of universal gravitation, we know that the force of gravity decreases with distance from the center of mass.
The formula for the force of gravity is F = G * (m1 * m2) / r^2, where:
- F is the force of gravity
- G is the gravitational constant (approximately 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- m1 and m2 are the masses of the two objects attracting each other (in this case, the astronaut and the earth)
- r is the distance between the centers of the two objects
When an object is 400 miles (or approximately 640 kilometers) above the surface of the earth, we need to find the new force of gravity acting on the astronaut.
First, we need to convert the 400 miles into meters, since the gravitational constant is in SI units (meters):
400 miles * 1.60934 kilometers/mile * 1000 meters/kilometer = 643,740 meters
Now, we can calculate the new force of gravity:
F' = G * (m1 * m2) / r'^2, where r' is the distance from the center of mass of the earth to the astronaut.
Let's assume the mass of the astronaut remains the same. So, m1 is constant.
Now, let's substitute the values into the equation:
F' = G * m1 * m2 / r'^2
F' = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (mass of the astronaut) / (643,740 meters)^2
Since we are given that the weight of the astronaut on the ground is 630 pounds, we need to convert it to kilograms:
630 pounds * 0.453592 kg/pound = 285.76 kg
Now we can calculate the weight of the astronaut at 400 miles above the surface:
F' = (6.67430 × 10^-11 m^3 kg^-1 s^-2) * (285.76 kg) / (643,740 meters)^2
Calculating this equation will give us the weight of the astronaut when they are 400 miles above the surface of the earth.
Regarding the astronauts being weightless, they are not actually weightless due to being far from the gravity of the earth. In fact, the force of gravity is still significant at that distance.
The reason astronauts experience weightlessness in orbit is because they are in free-fall around the earth. They are constantly falling towards the earth but moving with enough horizontal velocity that they keep missing the surface. This free-fall state creates the sensation of weightlessness since there is no normal force acting against them. They and their spacecraft are in a state of constant free-fall around the earth, giving the impression of being weightless.