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Jeb (3 kB) does some space travel and lands on Saturn $ moon Titan; which has a diameter of 116 000 km and mass 0f 5.68 X 1026 k8: a) What gravitational force does ...

Question

Jeb (3 kB) does some space travel and lands on Saturn $ moon Titan; which has a diameter of 116 000 km and mass 0f 5.68 X 1026 k8: a) What gravitational force does he feel? b) How many kilometers from the moon'$ surface is he. if he feels . gravitational pull 0f 32.7 N? For your answers below enter 3_significant digits ONLY (no units or directions herc; include them instead in full solutions). Ex: Actual answer: 12.035 You enter: 12.0 Actual answer: 1234 You enter: 1230 Your full solutions

Jeb (3 kB) does some space travel and lands on Saturn $ moon Titan; which has a diameter of 116 000 km and mass 0f 5.68 X 1026 k8: a) What gravitational force does he feel? b) How many kilometers from the moon'$ surface is he. if he feels . gravitational pull 0f 32.7 N? For your answers below enter 3_significant digits ONLY (no units or directions herc; include them instead in full solutions). Ex: Actual answer: 12.035 You enter: 12.0 Actual answer: 1234 You enter: 1230 Your full solutions should include diagram:



Answers

Figure 11.27 shows the variation with distance $r$ from the centre of a planet of the combined gravitational potential due to the planet (of mass $M$ ) and its moon (of mass $m$ ) along the line joining the planet and the moon. The horizontal axis is labelled $\frac{t}{d},$ where $d$ is the centre-to-centre separation of the planet and the moon. (a) The distance $d$ is equal to $4.8 \times 10^{8} \mathrm{m}$. Use the graph to calculate the magnitude of the gravitational field strength at the point where $\frac{t}{d}=0.20$. (b) Explain the physical significance of the point where $\frac{t}{d}=0.75$. (c) Using the graph, calculate the ratio $\frac{M}{m}$.

So here. In order to calculate the net force than that, don't gravitational force of the planets on the earth. We need to find the distance between these planets and the earth. So we can say that our some TV, which would be the distance between the Earth and Venus, would be 150 minus 108 times 10 to the six kilometers. And this is giving us 4.2 times 10 to the 10th meters. Ah, we can say the the distance between Earth and Jupiter would be equal to 778 minus 1 50 times 10 to the six kilometers. This is giving us 6.28 times 10 to the 11th meters. The distance between Earth and Saturn's would be equal to 1430 minus 150 times 10 to the six kilometers. And this is giving us 1.28 times 10 to the 12th meters. We can calculate the net force on the Earth by calculating the force of the gravitational force of Jupiter on the Earth. The gravitational force of Saturn on the earth minus the gravitational force of Venus on the earth. And that is because Venus would be technically if all of them are aligned with the sun, we can see the Venus would be on the other side of earth. So while Jupiter and Saturn would be closer to the outer solar system, we would be considering thie positive ex direction towards thie. Ah, out towards the end of the solar system, rather than towards the sun towards the sun would be considered the negative X direction. So we can say that this is going to be equal to the gravitational, constant times the mass of the earth squared and then times 318. This would be how many times more massive the planet is compared to Earth. So Jupiter has 318 times more mass than earth and then divided by the radius between Earth and Jupiter squared plus 95.1, divided by the radius between Earth and Saturn squared minus 0.815 divided by the distance between Earth and Venus squared my apologies. And so we can say that we can solve the net. Gravitational force on the earth will be 6.67 times 10 to the negative 11th times, 5.97 times 10 to the 24th kilograms Quantity squared and then this would be ah, 6.28 times 10 to the 11th meters squared on three eighteen 18 plus 95.1, given by 1.28 times 10 to the 12 meters quantity squared minus 0.815 divided by 4.2 times 10 to the 10th meters Quantity squared and we find that the net gravitational pull on the earth is going to be equal to 9.56 times 10 to the 17th Nunes. So this would be the answer to the first part. This would be the net gravitational pull of all the planets on earth. And then we want to find the ah net the gravitational pull of the sun on the Earth. Rather, we concise son on the earth and this would be equal to the gravitational, constant times the mass of the earth mass of the sun. But about the distance between the earth and the sun squared, this would be equal to 6.67 times 10 to the negative 11th times the mass of the Earth 5.97 times 10 to the 24th kilograms times 1.99 times 10 to the 30th kilograms on. Then this would be all divided by the radius of the rather the distance between the earth and the sun. And this is ah, 1.5 times 10 till the 11th meters quantity squared on. We find that the force of the sun on the earth would be equal Tio 3.552 times 10 to the 22nd Nunes s o Take note of this. The final part of the question asked us to find the ratio between these two. So the gravitational force of all the planets on earth, divided by the gravitational force of the sun on the earth, would be equal to 9.56 times 10 to the 17th divided by 3.552 times 10 to the 22nd and this is giving us 2.7 times 10 to the negative fifth. So essentially, this would be 27 millions. So then that gravitational force of all of all of the major planets on earth would only be 27 millions of the magnitude of the gravitational force of the sun on the earth. So extremely small. Yeah, the earth. The sun is, of course, what keeps us in our rotation. That is the end of the solution. Thank you for watching.

Everyone, This is question number forty one from chapter five. This problem talks about the planets lining up and then ask us to calculate the total force on the earth do to Venus, Jupiter and Saturn, assuming they're all foreign line were given the masses of each planet as compared to Earth's mass, the mean distance from the sun. And then we're asked, what fraction of the sun's force on earth is this? Okay, so we need to start this problem by finding out distances because we know that gravitational force on an object equals and one and two over r squared, which is the distance between the two. So we need to find the distances between each planet. So are eat a Venus equals Ah, let's see, that's gonna be one fifty minus one Oh, eight. Thank you. Times ten to the sixth, that is going to equal four point two times ten, four point two times ten to the ten meters and then our earth to Jupiter. That is seven, seventy eight minus one, fifty times ten to the six kilometers. And that equals six point two eight times ten to the eleventh meters. And then we have earth to Saturn, which is fourteen hundred and thirty minus one fifty times ten to the sixth kilometers. And that is equal to one point two eight times tended the twelve meters. Okay, okay, so now we can write out a full equation. And I wrote a little diagram appear, Sun Venus Earth, Jupiter, Saturn. And if we did not write to be positive than Venus is going to exert a leftward force force, and Jupiter and Saturn are going to exert a positive force. Okay, so let's write out the full expression going to go to the next page. So force on the earth is equal to G. Yeah, And me Earth em Jupiter over our e t j. And that's squared plus G and earth. I am Saturn over our he Teo s square minus G m e ems. And what's your last one? Venus over R TV squared. Okay, so now what? Plugging our numbers. So we're gonna have Essie equals. We can pull the gr We're gonna have a M E. Times three. Eighteen Emmy over the radius. I'm gonna leave the radius out for one more turn because, um, gonna simplify further here. Yeah, Any five point one at me. Oh, over the radius. Eat s squared minus M E zero point eight one five Emmy over Radius E to B squared. Okay, so my point writing that wear so we can take that m e squared out as well. And then we're left with a final expression at the equals. G m Earth squared times. Yeah. Three. Eighteen over six point two meters. Mom plus ninety five point one over one point two eight times ten to the twelfth. Mears. Oh, these Air square. Excuse me. Radius R squared minus zero point eight one five over four point two. Claire's ten to the ten. It's weird. Um, we can plug in G and M e squared as well. So, Effie, if will you do that entire right side of the question that I just rode out and then add G six point six seven times ten to the minus eleven new meters squared per kilogram times mass of the earth. Oh, kilogram squared to's mother mass of the earth. Five point nine seven times ten to the twenty four kilograms that squared times. When you do that expression on the right side to get four point O too. Times ten to the minus twenty two. No. All right. When you plug all that in and you get force on the earth do to these three planets is equal to nine point six times ten to the seventeen Newtons. Okay? And now we go on to the second part of the good question, Which the force of the earth on the sun. What fraction is that? So now we have F e due to the sun is equal to G M E m s over r E t s squared. And that is equal to six point six seven times. Tend the minus eleven noon meter squared per kilogram squared five point nine seven times ten to the twenty fourth kilograms mass of the earth mass of the sun, one point nine nine times ten to the thirty kilograms divided by the race between the two. One point five times ten to the eleven meters squared puggle of that end. And you get three point five two times ten of the twenty two Nunes and that is force of the sun on the earth. Okay, And now we compare the two eso to the ratio so force of planets on Earth over force of sun on Earth is equal. Tio nine point five six times ten to the seventeenth Newtons over three point five to times ten to the twenty two news. And that gives us a ratio of two point seven times ten to the minus seventh. So that means that the force of Venus, Jupiter and Saturn is twenty seven millions of the force that the sun exerts on the earth.

Pro Clinton. The universal in reputational constant is approximately 14. G is able to 6.67 time stamp or negative 11 m cube over kilograms, second square and the cinema. The semi major axis off on the earth is orchestra approximately is equal to 149.6 times 10.6 kilometers. So sensitivity is equal toe by game three over to over square road off GM. But people toe one year, which is 365 points and 24 36 years years seconds then one year, uh then and is equal to 45 square times 149.6 times 10 for six or 4 3/6 40.67 time simple negative. 11 3655 24 536 years. Here Square which opportunity for 1.99 times 10 for three for 30 kg

Question number four yeses to find the total force on Earth when the planets Venus, Jupiter and Saturn are in a line with it. Problem states that every 400 years four of the planets get in a line with each other and their gravitational forces affect one another. So let's draw this out on one side of the sun. We have Venus, Earth, Jupiter and Saturday set apart from the other. Despite during with the ring system. Now all four of these planets gravitationally affect one another in their orbits around the sun. The problem wants us toe determine the total force exerted upon Earth by the other three planets here. Now the problem gives us each planets mass. I'll write them all down here. The massive Venus M Savi is 0.815 a m sub e where ems of E is equal to one earth mass that so I'm denoting Earth's Mass there with him. So he and the mass of Jupiter, the largest planet, is equal to 318 times m sabih, while the massive Saturn is equal toe 95.1 m sabih were also given the average distance between each of these planets and the sun. The distance from Venus to the sun is 108 million kilometers are 108 times 10 to the six kilometers. As I'm writing here, Earth's distance from the sun is equal to 150 times tend to the six kilometers in the planet. Jupiter is 778 times 10 to the six kilometers finally Saturn. Why's it 1400 30 times tend to the six kilometers a little over a billion kilometers from the sun. Now, since we're trying to find the force exerted on Earth, we need to find each planet's distance from Earth here. The way we do that, IHS, we find the difference between each planet's distance from the sun and earth's distance from the sun. So the distance Venus is from Earth can be found by taking 150 minus 108 and multiplying by 10 to the six kilometers that gives us a value of 4.2 times. Tend to the seven kilometers or since we'll want to get everything in meters 4.2 times 10 to the 10 m. No, the distance between Earth and Jupiter is gonna be equal to 778 minus 150 times. Tend to the six kilometers, which turns out to be 6.28 times 10 to the eight kilometers. Getting that in 2 m, we have 6.28 times, 10 to the 11 m. Finally, we want to find the distance between Earth and Saturday that will be equal toe. 1430 minus 150 times. Tend to the six kilometers which comes out to be 1.28 times 10 to the nine kilometers or 1.28 times tend to the 12 m. Now each one of these planets lies at different distances from the sun and they will pull on each other in opposite directions. So Jupiter and Saturn life further from the sun than earth. So they'll exert a force on Earth opposite to that of Venus. Will say that the force exerted by Jupiter and Saturn is in the positive direction and the force exerted by Venus occurs in the negative direction. So to find the force exerted on Earth by Venus, Jupiter and Saturn, we need to take the gravitational constant times. The mass of earth times the mass of Jupiter over the distance between Earth and Jupiter squared plus the mass of Saturn divided by the distance between Earth and Saturn. Squared minus a massive Venus, divided by the distance between Venus and Earth squared. All right, let's plug in the values for the equations in parentheses and work our way out. From there we have the gravitational constant G times M. Savi well, plug in the values for those later multiplied by Jupiter's Mass. 318 times Imsa e divided by 6.28 times 10 to the 11 m squared distance between Earth and Jupiter, plus Saturn's Mass. 95.1 m So biggie, divided by 1.28 times 10 to the 12 m squared minus 0.815 times the mass of earth divided by 4.2 times, 10 to the 10 meters squared. Now we can plug in our values for the gravitational, constant and Earth's mass, and we can actually pull out another Earth Mass from inside the parentheses. Since we had all our planet's masses measured in terms of Earth's mass, so we'll have 6.67 times 10 to the minus 11 Newton's times meters squared over kilograms squared times 5.97 times 10 to the 24 kg squared times 4.2 times 10 to the minus 22 peters to the minus two. Yeah, meters squared is in the denominator in the part of the problem in parentheses, multiple calculations of numeric quantities. Later, we end up with the value of 9.56 times 10 to the 17 Newtons for the magnitude of force exerted on Earth by the planets Venus, Jupiter and Saturn. Now we're asked to compare this magnitude of force with that force which Sun exerts on the earth. To find the force exerted on Earth by the sun will need to do something similar to what we did previously. But with only two objects will need toe multiply the gravitational constant times the mass of the sun times the mass of earth divided by the distance between earth and the sun squared and that will be equal to our gravitational constant. 6.67 times 10 a minus 11 Newton's times meters squared over kilograms squared times sons mass of 1.99 times 10 to the 30 kg times Earth's mass of 5.97 times 10 to the 24 kg divided by 1.5 times 10 to the 11 m squared. Multiplying and dividing our way through that, we end up with a value for the force exerted on earth by the sun of 3.52 times 10 to the 22nd Newtons. Now let's see how the force exerted on Earth by Venus, Jupiter and Saturn compares with that of the force on Earth by the sun. Then divide the two. Our value for the force exerted on Earth by was other three planets waas 9.56 times 10 to the 17 Newtons and we divide that by 3.52 times 10 to the 22nd mutants. When we divide the two quantities, we find that the force exerted on earth by Venus stupider and Saturn is 2.72 times 10 to the minus five times that of the force exerted on Earth by the sun


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