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Your spaceship lands on an unknown planet. To determine the characteristics of this planet, you drop a 1.30 $\mathrm{kg}$ wrench from 5.00 $\mathrm{m}$ above the ground and measure that it hits the ground 0.811 s later. You also do enough surveying to determine that the circumference of the planet is $62,400 \mathrm{km}$ . (a) What is the mass of the planet, in kilograms? (b) Expressthe planet's mass in terms of the earth's mass.

(a) 2.25 $\times$ 10$^{25}$ kg (b) 3.76 $m_E$

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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Okay. So interesting question. Here you are lending on an unknown planet, and you want to do a bunch of experience. First thing you do, obviously you want to know that mass of the planet. So well, one of the one of the clues that she could extract from those experiments is that mystery conference or to buy our is 62 1004 100 kilometers, or that's right, it 6.24 Uh, I'm stem to seven years eso From this, we can extract the regis of this planet, which will be that number over to pie. Um, which haven't been numbers on, but I will beat it with you right now. So the regis 24 times into seven meters, divided by two pies is 9.9 tree times interest, six meters. Okay, so that will be useful to try to find the mass of the planet. Now, the way that I did this problem requires some physics from an earlier chapter. Um, Chapter two, actually, because, um, well, let's just talk about it when we get there. First thing, Um, first thing is, we want to know the mass. Right? So let's find an equation that links really? Yes. Which we know and the mass. And that is the question that we've been talking a lot in this chapter, which is if g gravitational force G times and one and two over our square. So, since we're at the surface of a planet, are in this equation is the same as the radius of the planet. Now, in one will be the mass of the planet on and to will be the mess off the wrench. And we know the mass of the wrench is one point. Treat kilograms. We know G. We knew our We do not know f g. And you want to know Mt. We do not know f G. However, we know that F g is equal to and may any force, this would be the sum of forces here. Um, applied on the wrench is always the mass times acceleration of the wrench. So that's very Gordon. Ah, and we should use him too, because we will be looking a drench. Strange has force f g. Um, actually, let me try it the other way because the planet pools on the wrench. So there you go. The ranch falls on the plant with Force ft. Okay, let's try to isolate. And one here then that gives us and won equals r squared over Jean. But we're still in trouble here because we still don't know a So this didn't really lead anywhere. Except now, using material from chapter two, we will find the value of a So in chapter two, there is, ah, very useful equation, which is that X equals x zero. Um, plus the original speed. I will explain the equation in more details after write it. Okay, so this isn't a question of cinematics. And basically, the way it goes is that given original position X zero, an original speed V zero X, we can calculate the position of an object that has an exploration Hey X, and we can calculate its position for any time. T So from the ranch experiment, we dropped a ranch five meters above the surface. I think that and we know it takes 0.811 seconds for drenched for that distance. So, um, regardless of what actually is x er eggs era, we know that X minus eggs era is equal to five meters and we know that t is recalled to 0.811 2nd So, in other words, the wrench has started from him. Speed of zero, because we just drop it and has traveled five years in 0.811 seconds. And this is due to this exploration provided by a planet. So that's, um, you're a X equals a soon I'll just use a equals. And, um, I'll, uh, i'll be rewriting this stop equation here, explain its exterior. Taking x zero here, Me on this side and then I can divide by 1/2. T squared. You guys like a in this first equation start here. All right, so now we have value for a that we can express in terms of this distance traveled, and this time it took to travel distance. Excellent. So let's let's combine those equations into one I have and one equals. Um, are too are screwed. Rather G times eight and eight is X my 60 divided by 1/2 t squared and so using our square using our equals 9.9 treat times 10 to 6 years and G equals 6.674 instant T minus 11 new tunes Meter, square over Kilograms square and X minus zero. As we said here is fighting years and time is zero point 811 seconds. Okay, so no more unknown variables here. They're all numbers. We can crunch numbers down and the result is 2.25 times 10 to 25 kilograms. And that is the answer to A Isn't Andi be? This one is shorter. Be just ask you to express this ratio as in terms of Earth's mess. And we knew Earth's mess is five point 97 times 10 to 24 programs. So that if we write, um, if we write 2.25 times 10 to the 25 program's over 5.97 times tend t four kilograms, it simply gives us treat 0.76 So the ratio she 0.76 are in other words, um, this new planet that we just landed on is 3.76 times heavier than earth. And this computes this

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