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.