The Near Earth Asteroid Rendezvous (NEAR) spacecraft, after tr...

Question

The Near Earth Asteroid Rendezvous (NEAR) spacecraft, after traveling 2.1 billion km, orbited the asteroid Eros with an orbital radius of about 20 km. Eros is roughly 40km x 6km x 6km. Assume Eros has a density (mass/volume) of about 2.3 x 10³ kg/m³ .

(b) What would g be at the surface of a spherical Eros?

(b) What would g be at the surface of a spherical Eros?

Accepted Answer

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use. In order to solve this problem. A space agency is preparing to send a mission crew to visit a planetoid on a spaceship. The spaceship is planned to travel to the planetoid that is located 7.9 billion kilometers away and orbit it at an altitude of 15 kilometers above its surface given the dimensions of the planetoid as 25 kilometers by 31 kilometers by 35 kilometers and its density as 2.3 multiplied by 10 to the power of 3 kg per meter cubed estimate what the value of the gravitational force G is at the Plano's surface to simplify our calculations. Let us treat the planetoid like a sphere which has the same density and volume. Note that the gravitational constant is G capital G is equal to 6.67 multiplied by 10 to the power of negative 11 Newton's multiplied by meters squared per kilograms squared. So that's our end goal. Our end goal is we're trying to estimate what the value of the gravitational force G is at the planetoid surface. So that's our final answer that we're ultimately trying to solve for is what is the value of lower case G. So looking at our multiple choice answers, let us note that all of them are in the same units of meters per second squared. So let's read them off to see what our final answer might be. So A is 1.2 multiplied by 10 to the power of negative two, B is 6.7 multiplied by 10 to the power of negative 14 C is 3.1 multiplied by 10 to the power of negative 20 and D is 9.9 multiplied by 10 to the power of negative 20. OK. So first off, we will first need to solve for the radius of the planetoid noting that it is a sphere with the same density and volume. Let us recall and note that the mass capital M is equal to the density row multiplied by V the volume. Awesome. Where row of course is our density and V is the volume. And once again, capital M is our mass. So since the density and volume is the same, the mass will also be the same. Therefore, this sphere will have the same mass as the planetoid note, since the sphere and the planetoid have the same volume, we can determine the radius of the planetoid by setting the two volume equations equal to each other. Like so and let us quickly recall that the volume of a sphere equation two. So let us recall that the volume of a sphere is V is equal to four thirds multiplied by pi multiplied by R cube, where R is the radius of the planetoid V is the volume of the planetoid. And now we need to rearrange this equation to help us isolate and solve for R. So let's call this equation one equation one states that R the radius is equal to three multiplied by V. So three multiplied by V divided by four, multiplied by pi all to the power of one third. So now we must recall and use the gravitational acceleration equation and let us call this equation two an equation. Two states that lowercase G is equal to capital G multiplied by capital M all divided by R square where once again capital G is the gravitational constant. So now we need to plug in the value of capital R into equation two to get. So we can say that G is equal to and be specific lowercase G is equal to capital G multiplied by capital M divided by three, multiplied by V, divided by four multiplied by pi and this is to the power of one third. And then this expression is all to the power of two and then this is equal to. So now what we're trying to do is we're trying to simplify down as far as we can go, we were trying to simplify our. So we're out here now we're trying to simplify the gravitational acceleration equation down to as simplified as we can get it. So now we are taking capital G multiplied by capital M divided by parenthesis, parentheses, three multiplied by V divided by four multiplied by pi to the power of two all to the power of one third. And this is all equal to because we're still simplifying parentheses, 16 multiplied by pi squared to the power of one third multiplied by capital G multiplied by M specifically capital M divided by nine multiplied by V squared all to the power of one third. Awesome. We're moving right along here. So now at this stage, we need to use the mass equation which we wrote above as capital M equals row multiplied by V. So we need to plug this into this equation. So I'll write a little arrow here and say that lowercase G is equal to 16 multiplied by pi squared to the power of one third multiplied by capital G multiplied by row. So row, where row, once again is our density multiplied by V divided by can the C nine multiplied by V squared to the power of one third. And we can simplify this equation. So when we simplify, we will get, so I'm gonna skip a step here. Since you're getting the idea, we have to use a bunch of algebra to rearrange and simplify. So we'll know that to the, the most simplified form, you will find that G lowercase G is equal to. So using algebra, the most simplified form is 16 multiplied by pi squared to the power of one third multiplied by capital G multiplied by row divided by nine to the power of one third. And then this whole expression will be multiplied by V to the power of three, divided by V to the power of two all to the power of one third. Awesome. So now if we simplify one last time, which I said that was our most simplified term. Oops I lied. We can simplify this one last time and we can say 16 multiplied by pi squared multiplied by V divided by nine all to the power of one third multiplied by capital G multiplied by row. OK. So now we can plug in all of our known variables to sulfur G, which is our final answer. So we will find that lowercase G is equal to parentheses, 16 multiplied by pi square multiplied by 25 kilometers. But we need to use dimensional analysis to convert 25 kilometers to meters. So we need to multiply this by, we need to recall that there's 1000 meters in one ameer and we need to do the same thing for 31 kilometers. You need to multiply it by 1000 m and one kilometer. And we need to multiply this by 30 five kilometers. Awesome. 35 kilometers. Well, and don't forget your units, 35 kilometers multiplied by 1000 meters and one kilometer. Awesome. And don't forget that this is all divided by nine. And don't forget that this whole expression is to the power of one third. Awesome. So now don't forget that we have to multiply that whole expression by our gravitational constant, which is 6.67 multiplied by 10 to the power of negative 11 in its units are newtons multiplied by meters squared. And here I'm gonna move. So, so we don't forget. So when I said the whole expression is to the power of one third, one third, that's very important multiplied by our gravitational constant, which is newtons multiplied by meters squared per kilograms squared for our units. And then we need to multiply it by our density which is given to us as 2.3 multiplied by 10 to the power of. So we need to take 10 to the power of three kilograms per meters. Cute. OK. Fantastic. So let me plug that indoor calculator. We will get that G is equal to 1.19 seven multiplied by 10 to the power of negative 2 m per second squared, which is approximately equal to 1.2 multiplied by 10 to the power of negative 2 m per second squared. Awesome. And that's it. That's our final answer. Hooray, we did it. And this is when we round our final answer to two significant figures. So looking at our multiple choice answers, the correct answer has to be the letter A 1.2 multiplied by 10 to the power of negative 2 m per second squared. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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