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Question: 

You are explaining to friends why an astronaut feels weightless orbiting in the space shuttle, and they respond that they thought gravity was just a lot weaker up there. Convince them that it isn’t so by calculating how much weaker (in %) gravity is 380 km above the Earth’s surface.

Source: Giancoli, Douglas C., Physics: Principles with Applications, 7th Edition, 2014.

Quick Answer: 

$11\% \textrm{ weaker}$

Giancoli 7th Edition, Chapter 5, Problem 40

(3:11)

Chapter 5, Problem 40 is solved.

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Transcript for this Giancoli solution

This is Giancoli Answers with Mr. Dychko. The acceleration due to gravity in orbit is capital G times mass of the Earth divided by the distance from the orbit position to the center of the Earth, squared. And so, this radius of orbit is gonna be radius of the Earth plus the distance above the surface of the Earth, which is 380 kilometers, which I write as 380 times 10 to the 3 meters. So, the total distance would be 6.38 times 10 to the 6 meters plus 380 times 10 to the 3, which is 6.76 times 10 to the 6 meters. And so, we put that in the denominator of this acceleration due to gravity formula; square it, and on the top, we are multiplying by— the gravitational constant,— 6.67 times 10 to the minus 11 and then times—mass of the Earth— 5.98 times 10 to the 24 kilograms. And then we take this result of 8.7284 and find the difference between that and the acceleration due to gravity on the surface of the Earth, divide by g E, and this gives us the percent by which the acceleration due to gravity is weaker. So, the amount that it's weaker in absolute terms divided by the original amount gives you percent difference. And on a calculator, it looks like that. So, when you are explaining to your friend why the gravity is still acting, even though, significantly still acting, it's only 11 percent weaker up there in orbit, but certainly, the astronaut experiences a very different type of gravity. They experience a perceived weightlessness as though there was no gravity because they are constantly falling. A person, in orbit, is going around the Earth here, and that's the Earth say and here's their ship and as they are moving forwards in this ship, the ship is at the same time, falling down, towards the center of the Earth. And so in a state of constant falling, there's perceived weightlessness. And so here's a picture of somebody, you know, just jumping out of a plane about to open their parachute but they haven't done it yet and they are falling and there's no normal force on them; this normal force is really what you experience with gravity. And when you are in free fall, there's no normal force and so there's no perception of gravity; same with the jumping off a cliff and diving into some water. There's no normal force, you get the butterfly feeling. But gravity is still acting on this person, clearly, because they are falling and there is gravity still acting on this person because they are falling as well and in the same way, there's gravity acting on this astronaut because the astronaut is falling. It just doesn't quite look the same since they are also moving to the side at the same time but, nevertheless, it's the same idea; they are still falling. And, gravity is still acting and it's responsible for that falling and because there's no perception of normal force, there's a perceived weightlessness.