Gravitational Potential Energy for Systems of Masses
8. Centripetal Forces & Gravitation
Gravitational Potential Energy for Systems of Masses - Video Tutorials & Practice Problems
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concept
Gravitational Potential Energy for Systems of Masses
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Hey guys. So in the last couple of videos, we saw that any two masses apart from each other have some gravitational potential energy. Well, in this video, you're gonna see what happens when you have three or four masses in a system. All right, let's go ahead and check it out. So the idea is that if I take a look at this triangle that I got over here, I have three masses, right? And I know that between masses one and two, there's some distance in between them. So I could figure out what the gravitational potential energy is. And I'm gonna call that U 12 because it's between masses one and two. But if you take a look, this isn't just the only pair of masses that I have in this diagram because I also have a pair between mass one and mass three and I have another pair between mass two and M three as well. So the idea is that all of these pairs of masses actually have energies between them. So I have U 23 over here and I've got U 13. So if I were to be asked on a problem. What's the total amount of gravitational potential energy of this whole system at masses? All I have to do is just simply add the energies for each individual pair. And the reason I can do this is because energies unlike forces which are vectors are scalers, so we don't have to deal with any signs or cosines or vector decomposition or trigger or anything like that. We can just simply add these numbers together for each of these pairs because scalers are just numbers. So in general, if I have a system of masses, all I have to do is just add the energies U 12, U, 13, U 23 and so on and so forth. Depending on how many number of masses I have. Usually you'll see it just about three or four in a given problem. All right. So let's just say I magically knew what all of these masses were and the distances between them, I'm gonna just make up some numbers here. Let's say I knew that the potential energy between these masses was negative two joules and I have this was negative three joules and negative four joules. And if I were to be asked what the total gravitational potential energy is, the sum of all of the gravitational potential energies is just gonna be adding the those negative numbers up. So I'm gonna have negative two plus negative three plus negative four. And so if you add all of these things up together, you're gonna get negative nine jewels as your total gravitational potential energy. All right. So it's very, very straightforward. Let's go ahead and just jump into a real example. So we're being asked to calculate what the gravitational potential energy is in this equilateral triangle system of masses that have. So you've got these 10 kg and 20 kg masses that are all positioned in an equilateral triangle. Now, if we want to figure out what the total gravitational potential energy is. So in other words, Sigma ug that Sigma just means the sum, all you have to do is just add all of these things up together. Now, what I'd like to do in these kinds of problems is I like to label my masses. So I'd like to label them just so I don't lose track. I've got this is M one M two and this is gonna be M three. OK. So I've got U 12, U 13 and then U 23. So I'm just gonna go ahead and write in the those individually out. So I've got the total gravitational potential energy is negative G the product of mass one mass two. And then I've got GM one M three and now I've got negative GM two M three. And you've got to remember that these negative signs are here because you're adding together these potential energies which are negative. OK. So there's always negative signs right there. All right. Now, the only thing we have to do is just figure out what the distances between each of these things are. Now, we're told this is actually an equilateral triangle. Now, what that means is that all of these distances, 60 centimeters are the same between all of these points right here, which is actually really nice because it ends up simplifying things for us. So that means that all of these distances right here, I don't have to go find them individually. They're all just equal to 60 centimeters, which is nice for me. So all we have to do is just plug in the masses and just the GS and then just divide by that 60 centimeters and that's it. So that means the total gravitational potential energy is negative G. Now, I'm not gonna keep on plugging this 6.67 times 10 of the minus 11. You guys can do that and just make sure that you get the right answer. So I'm gonna have negative G. Now, I've got to just plug in the masses. I've got 10 and 10 and then divided by 60 centimeters, which is 0.6. Now minus G, now I've got 10 and 20 divided by 0.6 minus G, 1020 divided by 0.6. OK. And you can just go ahead and add all of these things up together. Uh One thing you could do is you could just say that this 20 is exactly twice of what this number is. And all these other numbers are the same. So you can kind of use shortcuts like that to make the math a little bit easier. And you should get the total gravitational potential is negative 5.55 times 10 to the minus. And that's gonna be eight in Joules. All right. And that's our final answer. Let me know if you guys have any questions with this and let's go ahead and get some more practice.
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Problem
Problem
What is the total gravitational potential energy of this system of masses?
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example
Energy Conservation in Three-Mass System
Video duration:
8m
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Hey guys, let's work out this example together. So in this first part, we're supposed to figure out what the total gravitational potential energy is of the system. So to do that, I basically have to have all of my masses labeled and I need all the distances. So fortunately I have that. So the total gravitational potential energy is gonna be the sum of the potential energies between all of these masses right here. So I've got U 12 U 23 plus U 13, right? So I can go ahead and start basically writing out all of those expressions U 12 is gonna be negative G and this is gonna be mass one mass two divided by R 12. And that then I've got G M two M three over R 23 minus GM one M three over R 13. So let's just make sure that I have all these variables before I start plugging stuff in, I might need to go off and figure something out. So I've got the masses of all of these things in these diagrams. And then I also have the distances between them, the 50 50 60 centimeters. So that means I actually have everything I need to start plugging in to these formulas. So let's go ahead and start doing that. So we've got the total gravitational potential is gonna be and I've got negative G, I'm just gonna keep this as a letter. The number is right here. I just don't wanna have to write it out over and over again. And then I've got 25 and then 10 divided by the distance between them, 0.5 because we have 50 centimeters and we've got minus G 10 and 25 divided by 0.5. Notice that these two numbers are actually the same 25 times 10 is the same thing as 10 times 25. So both of these things basically, you just need to plug it into your calculator once and then multiply it by two. And then we've got minus G and then we got the product of 2525 divided by the distance between M one and M three, which is 60 centimeters, not 50. So you got a 0.6. If you go ahead and start working this stuff out again, just plug this in once and multiply it by two, you're gonna get, this is negative 6.68 times 10 to the minus eight. And then this guy over here is negative 6.95 times 10 to the minus eight. We have to subtract him because again, everything is subtracted right here. Right. Great. So if we go ahead and plug that stuff in, we get that the total gravitational potential energy is equal to negative 1.36 times 10 to the minus seven. And that's joules great. So that's the first part again, flexing that muscle trying to figure out what the gravitational potential energy is. And the second part, let's see what happens. The 2 25 kg masses are fixed, right? So they don't move and cannot move. And then this M two right here is gonna be released from rest. And as it's released from rest, it's gonna be pulled directly towards the center because there's symmetry, you have the same masses. Exactly. And do we have the um symmetrical distance? So it's gonna be pulled in direction and we're supposed to figure out what the velocity is going to be when it reaches this point and passes through the center. So that's our target, variable V final. So in this part B right here, first, we're gonna start from our V final. And if we're talking about velocities and we're talking about um gravitation, we have to use energy conservation. So we're gonna use energy conservation. So let's write out the equation we've got kinetic initial plus potential, initial plus any work that's done by non conserv forces equals final kinetic and final potential. The one thing that we have to remember though is that because we're working with multiple masses right here, we have to use energy conservation for the entire system for all three masses. So we actually basically have to see if anything has any kinetic energy, any potential energy. We have to look at the entire system as a whole. So let's go ahead and do that. We're told that all of these objects are fixed or are released from rest. So that means that there nothing is moving in the beginning. So it means all the kinetic energy is equal to zero, there's nothing moving, there is some gravitational potential. And that's actually what we figured out in part. A. Uh let me go ahead and scoot this down a little bit actually just so it doesn't get. And let's see, once we release these objects, the only force that acts on them is gravity. So there's no work done by any non conserv forces. Now, the final kinetic energy is gonna come from the fact that anything is moving in the system after it's some final state, we know that this mass two when it gets towards the middle is gonna be moving off in this direction. So there is gonna be some kinetic energy, but these two objects are fixed, which means that they don't. So the only thing that contributes some kinetic energy is gonna be one half M two V final squared. And that's what our target variable is. So that's our V final. And then we have the uh final potential energies in which there's gonna be some because there's gonna be some distances involved, right? So there's always potential energy. Great. So let's go ahead and start writing out what those formulas are. The initial gravitational potential is gonna be just the U 12, initial U 23 initial and then U 13 initial and then the final gravitational potential energy is going to be U 12, final, U 23, final and then U 13 final. Now, the reason again, I had to put finals and initials is because this object moves this M two moves towards the center. And as that happens, these distances the 50 centimeters will change, they'll go from 50. And then finally, over here, it'll be 3030 over here. So because there's distances that are changing, our gravitational potential energies are going to be changing. So what we can do is we can basically figure out uh we can basically write out all of these expressions right here, right? So you've got, again, this is just gonna be uh I'm just gonna go ahead and copy this guy right over here and then move that down over here. We know that this is the expression for all of the gravitational potential energies. And this is gonna be a minus sign right here. And then we have to write M one half MV final squared and then minus GM one M two over R 12, final minus GM two M three over R 23, final minus GM one M three over R 13 final. So I know this looks like a huge expression. But we can actually um use some, we can basically look at this diagram and see if we can eliminate some of these terms from this equation. So we know what these, these distances actually are all of the RS in the initial state. And we actually have a number uh for these guys, these are just going to simplify down to negative 6.68 times 10 to the minus eight. We actually figured that out before. Now, let's look at, look at these two terms, the masses, 13 and mass 13 final. So the masses between one and three and the distances between them actually don't change as M two gets closer. We can see that because here they're separated by 60 centimeters. And then when this thing gets finally close to the center, these things are fixed in place. So they still remain at 60 centimeters. So in other words, these distances are actually the same. So both of those numbers, when you actually work that out are going to be the same. So we can actually um basically cancel those out and then keep going with our equation. So we've got one half MV final squared and then this is gonna be negative G. Now we've got M one and M two that's gonna be 25 and 10 divided by the final distance between them. If you look back at the diagram, now, they're gonna be 30 centimeters apart. So we're gonna write 0.3 and it's gonna be the same exact thing G 1025 divided by 0.3. So if you work this out, these two together again, multiple, just, you know, calculate one and then multiplied by two, we're gonna get negative 6.68 times 10 to the minus eight equals one half. And then the mass of this guy is gonna be 10, right? That's M two equals oh sorry, that's gonna be minus uh And then the term on the right becomes negative 1.12 times 10 to the minus seven. So we're gonna move this guy over and, and uh and add, and this term on the left becomes 04.52 times 10 to the negative eight. And it's a positive number equals one half of 10. And this is gonna be a V final squared. So now we all, we all we have to do is basically uh multiply by one half and divide by 10, which is basically like dividing by five. So when you move all this stuff over, this is just gonna be 9.04 times 10 to the minus nine and that's gonna be equal to V final squared. So the last thing you have to do is just take the square root and we get that V final. Uh When you take the square roots is gonna be 9.51 times 10 to the minus 5 m per second, that's equal to V final. So let me know if you guys have any questions with that. It's very important that when you do energy conservation, that you have to do energy conservation of the entire system and then see what you can cancel out. Anyway, let me know if you have, if you have any questions.
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