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Exploring Europa. Europa, a satellite of Jupiter, is believed to have an oceanof liquid water (with the possibility of life ) beneath its icy surface. (See Figure 6.29 . $)$ Europa is 3130 $\mathrm{km}$ in diameter and has a mass of $4.78 \times 10^{22} \mathrm{kg} .$ In the future, we will surely want to send astronauts to investigate Europa. In planning such a future mission, what is thefastest that such an astronaut could walk on the surface of Europa if her legs are 1.0 $\mathrm{m}$ long?

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Physics 101 Mechanics

Chapter 6

Circular Motion and Gravitatio

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Applying Newton's Laws

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So in this question, we are presented with a scenario where scientists would want to test a lander under the same gravity conditions as the surface of Europa. And we're told that one way that we can do this is to rotate the lander at the end of an arm, Um, the arm being 4.25 m and were asked to calculate at what angular speed we would need to rotate this arm in order to create the same acceleration. That, um, would be the acceleration due to gravity on the surface of Europa. And then we're given some astronomical data about, um, Europa itself. So in order to do this, what we want to do is first to determine what the acceleration due to gravity is on the surface of Europa, and then we can calculate a corresponding angular speed from that. So let's do that's we've got, um the acceleration due to gravity is gonna be big G times the mass of Europa, divided by its radius squared. And so what we'll do is we'll simply plug in those values that were given. So the mass is 4.8 times. Tend to the 22 kg and just be careful because we are given the diameter, and we, of course, want to use the radius. So we're gonna divide that by two. So the, um the radius is 1569 kilometers, and so we're gonna use 1.569 times tended to six meters here. And so once we go ahead and plug that into a calculator would get a value of about 1.3 m per second squared for the acceleration due to gravity. And so this is the acceleration, the centripetal acceleration that we would want to create on the rotating arm. So I'm going to write that down as sort of a given. We want thesis in triple acceleration created by the rotating arm to be 1.3 m per second squared. And if you remember something about, um, circular motion, then you'll know that the acceleration, the centripetal acceleration, that an object moving in a circle experiences is equal to the speed squared divided by the radius of the circle that is being executed. Now, we are not interested in this this speed. We're interested in the angular speed, so this sort of translational speed is equal to our times angular speed. And that is what we're interested in. So we can rewrite the acceleration. Ah, in terms of Omega, just buy something that in one of the arts will cancel and we'll get our Omega squared. So in order to calculate Omega, I'm just gonna divide by, are on both sides and take the square root. And so we get a final formula here for Omega, um, the square root of acceleration who divided by the radius. So the acceleration is one points three and the radius is the radius of the or the length of the arm that were given right, cause that's the radius of the circle that the lander would execute. And so we get a value for the angular speed of 0.55 And this is in terms of radiance per second because we've been using standard units up until this point. So in order to, um, get this into RPM, which is what we were asked to find, we need to do a unit conversion, some risk energy that hear him. So instead of radiance, I'd like to use rotations. So there are, um two pi radiance in one rotation. Right in one circle, there are to pry radiance and I want to get rid of seconds. So I need to multiply by another conversion factor that has seconds in the top. And I'm replacing that by minutes. So we'll put that in the bottom. And so to do the conversion, we just need to divide by two pi and multiply by 60. Once we do that, we get 5.28 and that is now in rotations per minute or rpm, as we commonly say. So this is the final answer here, and this is how fast we would need to rotate that arm so create the X same acceleration as the acceleration due to gravity at the surface of Europa.

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