Jupiter is about 320 times as massive as the Earth. Thus, it h...

Question

Jupiter is about 320 times as massive as the Earth. Thus, it has been claimed that a person would be crushed by the force of gravity on a planet the size of Jupiter because people cannot survive more than a few g’s. Calculate the number of g’s a person would experience at Jupiter’s equator, using the following data for Jupiter: mass = 1.9 x 10²⁷ kg, equatorial radius = 7.1 x 10⁴km, rotation period = 9 hr 55 mins. Take the centripetal acceleration into account.

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 science fiction author is considering the possibility of sending a monkey off to another planet. Considering the centripetal acceleration determine how many GS would the monkey experience at the equator of the planet? Given that the mass of the planet is equal to 2.9 multiplied by 10 to the power of 27 kg. The radius of the planet at the equator is equal to 4.5 multiplied by 10 to the power of four kilometers. The length of a day on the planet is equal to seven hours and 33 minutes. So that's our goal. Our end goal is we're trying to figure out how many Gs would the monkey experience at this particular planet that he's being sent off to? And that's our final answer that we're ultimately looking for. So looking at our multiple choice answers, we have some value multiplied by the unit of GS where G is the gravitational acceleration for that particular planet. So A is equal to 9.5 B is equal to 9.8 C is equal to 93 and D is equal to 4700. Awesome. So first off, let us note that at the equator, the monkey would essentially be in uniform circular motion of radius R which is the radius of the planet at the equator. Let us also note that capital N is the normal force that would be exerted on the monkey by the planet. So to better help us visualize this problem or to visualize all the forces at work in this particular prong to be exact, let us create a free body diagram of this particular prong. So as we could see, we have our planet which is represented by a sphere and then we have our monkey standing at the equator on our sphere, which the equator is the center of our sphere. We have the normal force pointing up and away from the monkey and it's some vector N is our normal force. And then we have the weight force of the monkey pointing towards the center of the planet which the weight forces the mass of the monkey multiplied by the acceleration due to gravity, which is vector G. So now that we have a visualization of what's happening in this particular prompt, let us also note that capital N is the apparent weight of the monkey on the planet. So if we divide the normal force by the mass of the monkey M, we will get capital N divided by M which will give us the acceleration value that the monkey would experience. Since the monkey would rotate about the center of the planet, the net force would be the centripetal force pointing towards the center of the planet. Thus, we can write and let's call this equation one, we can write that the sum of F is equal to the mass multiplied by gravity or the acceleration due to gravity. And we're gonna call this J I, sorry J sorry G subscript J minus capital N which is all equal to M multiplied by V squared divided by R. So once again, let's sit equation one is the sum of F which is equal to M multiplied by G subscript J minus capital N which is all equal to M multiplied by V squared divided by R where GJ is the gravitational acceleration on the planet. V is the linear speed of the monkey and R is the radius of the trajectory or in this case, it is the radius of the equator for this mystery planet. So now we need to take the direction towards the center of the planet to be the positive direction and the direction of the normal force to be negative. Thus, we can write and let's call this equation too that the normal force. Capital N is equal to M multiplied by GJ minus V squared divided by R. So let us note that GJ could be written as so GJ is equal to the gravitational force constant multiplied by capital M or capital M is the mass of the planet divided by R squared. So now we need to plug in the value of GJ into equation two to get that capital N is equal to M multiplied by capital G multiplied by capital M divided by our square minus V squared divided by R. OK. Awesome. So let us note once again that capital M is the mass of the planet. So now we need to write the equation for the apparent acceleration experienced by the monkey. And let us call this value G A. So this is the apparent acceleration. So let's call this equation three. So G A, the apparent acceleration is equal to capital N divided by lowercase M which is all equal to capital G. So capital G multiplied by capital M divided by R squared. So R squared minus V squared divided by R, let us also note that and let's call this equation four. That V is equal to two multiplied by pi multiplied by R all divided by capital T. Capital T is the period. So now at this point, we need to plug in the value of V into equation three and we can call this equation five. And equation five states that G A is equal to capital G multiplied by capital M divided by R squared minus four, multiplied by pi squared multiplied by R all divided by capital T two. Awesome. So now let us note that we are given the mass of the planet. And let's write this as a side note in blue that the mass of the planet capital M is equal to 2.9 multiplied by 10 to the power of 27 kilograms. We're also told that the radius of the planet at the equator is equal to R and it's equal to 4.5 multiplied by 10 to the power of four kilometers. So we need to write this down. So we need to also convert kilometers to meters. So we can take 4.5 and multiply it by 10 to the power of seven to convert it to meters. So I've gone ahead and did the conversion for you. But you can use dimensional analysis to do this convert conversion or you could quickly look it up. I've just given it to you to save some time. So now moving right along here, we now need to figure out the orbital period in units of seconds in order to help find our final answer in GS. So let us note that the orbital period T. So now we're gonna be using dimensional analysis to solve for T in seconds. So the time T or the period of our particular planet is equal to seven hours and 33 minutes. So it's equal to. So now we could start doing our conversion. So in seven hours, there is 3600 seconds in one hour and we need to add this to 33 minutes. So in 33 minutes, there is as we should be able to recall 60 seconds in one minute. So as you can see all of our units cancel out, but seconds, which is what we want our period to be in, We want it to be in seconds. So we plug that into our calculator. We will get 27180 seconds. So 27,180 seconds. So now we need to plug in these values back into equation five to solve for the value of G A which is our final answer. So G A is equal to our gravitational constant, which is numerically equal to 6.67 multiplied by 10 to the power of negative 11. Its units are newtons multiplied by meters squared, divided by kilograms. OK? Multiplied by 2.9 multiplied by 10 to the power of 27 kilograms. Awesome. And this is all divided by 4.5 multiplied by 10 to the power of seven meters squared minus or multiplied by pi squared, multiplied by 4.5 multiplied by 10 to the power of seven. So 10 multiplied by the power of seven meters divided by our second squared means it's our period squared. So 27,180 seconds squared. So let me plug that into our calculator. We will get that G A is equal to 93.1162 meters per second squared. OK? So now we need to convert meters per second squared to G's units. So we need to take G A which equals 93.1162 m per second squared. And we need to multiply so that we're gonna have to use our conversion factor. So we need to multiply this value by one divided by nine 0.81 meters per second square is G is like the acceleration due to gravity on earth. So we're comparing it to that. So when we plug this into our calculator rounding to one decimal place, we will get 9.5. Jeez. And that is our final answer. Hooray. We did it. So looking at our multiple choice answers, the correct answer has to be the letter A 9.5 G's. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

Key Concepts

Gravitational Force

Centripetal Acceleration

Acceleration due to Gravity

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