You are explaining to friends why astronauts feel weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there. Convince them and yourself that it isn't so by calculating how much weaker gravity is at h = 330 km above the Earth's surface.
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To calculate the strength of gravity at a specific height above the Earth's surface, we can use the equation for gravitational force. The force of gravity acting on an object is given by the equation:
F = G * (m1 * m2) / r^2
Where:
- F is the gravitational force
- G is the universal gravitational constant (approximately 6.67 x 10^-11 N m^2 / kg^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 (the Earth's radius plus the height above the surface)
In this case, we are interested in finding the ratio of the gravitational force experienced by the astronaut at 330 km above the Earth's surface compared to the force experienced on the surface of the Earth.
Step 1: Calculate the distance from the center of the Earth to the astronaut's position:
First, we need to convert the given height above the Earth's surface (330 km) to the total distance from the Earth's center. The average radius of the Earth is approximately 6,371 km.
Distance from center of Earth = Earth's radius + height above surface
Distance from center of Earth = 6,371 km + 330 km
Distance from center of Earth = 6,701 km
Step 2: Calculate the ratio of gravitational forces:
Now that we have the distance from the center of the Earth, we can calculate the ratio of the gravitational forces:
F at 330 km above the surface = G * (m1 * m2) / r^2
F on the surface of the Earth = G * (m1 * m2) / (Earth's radius)^2
Ratio of gravitational forces = F at 330 km above the surface / F on the surface of the Earth
Substituting the values into the equation, we get:
Ratio of gravitational forces = (G * (m1 * m2) / r^2) / (G * (m1 * m2) / (Earth's radius)^2)
Ratio of gravitational forces = (r^2 / (Earth's radius)^2)
Ratio of gravitational forces = (6701 km)^2 / (6371 km)^2
Calculating the above equation will give us the ratio of gravitational forces at a height 330 km above the Earth's surface compared to the force on the surface.
When we calculate the above expression, we find that the ratio of gravitational forces is approximately 0.89.
This means that the gravitational force experienced by an astronaut at 330 km above the Earth's surface is about 89% of the force experienced on the surface. So, gravity is slightly weaker at that height, but not significantly weaker.
Therefore, it is not the weakened gravity that causes the feeling of weightlessness for astronauts in orbit around the Earth, but rather the continuous freefall they experience as they orbit the Earth, where the centrifugal force of their orbital motion precisely balances the force of gravity.