10. Conservation of Energy
Motion Along Curved Paths
Curved Paths & Energy Conservation
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Hey guys, So for this video, I want to introduce you to a type of problem. I like to call the curved path problem. Let's check out the example and we'll come back here and fill out the rest. So I've got this block that is traveling with some speed and I want to figure out what is the minimum speed? What's this minimum V here? So that this block can travel up this curvy path and reached the top of the hill, just barely reached the top. So how do we solve this? So we have the heights of the hill, we know this height is equal to 20. So how would I solve this? Do I use forces? Do I use schematics? There's a couple of reasons why those approaches are going to work for one. We actually don't have the angle of the slope because it's constantly changing all the time. And even if we did actually know the angle, what's going to happen is imagine you had a box that was sort of at this point right here, the MG would point downwards and your component of MG that's pulling you down would point in this direction. But then when you reach the steeper part right here, the components pulling you down is actually in a different direction. So along the curvy path, your acceleration is never going to be constant. You can't you can't use forces or cinematics. So what I want to point out here is that you're always going to solve these curvy path problems by using energy conservation, there's no other way that you can solve these kinds of problems. Now, the reason I say only is because there's technically in some very rare cases you may be able to solve these using calculus, but almost always you're going to use them using energy, you're gonna solve them by using energy conservation. So let's go ahead and get to this, right? So we have our diagram here basically we're trying to figure out from point A to point B. What is the speed that we need? So we can reach the top of this 20 metre hill. So now we're just gonna go ahead and write our energy conservation equation. So this is going to be K. Initial plus you initial plus. Work done by non conservative equals K. Final. That's K. B. Plus you Final. So let's go ahead and eliminate and expand all the terms. So we do have some kinetic energy initial. That's really just gonna be related to this speed here. So we do uh do we have any gravitational potential? We'll hear what happens is if your height is 20 and you can say that your heights initial is equal to zero when you're at the ground here. So you don't have any gravitational potential. So do we have any work done by non conservative? Remember that is done that's work done to applied forces or play or work done by you or work done by friction. And you actually have neither one of those because you're basically just this block is traveling and there's gonna be no friction because it's on a smooth hill. So there's gonna be no work done by non conservative forces. Now, do we have any kinetic energy? Final? The whole idea is that you might be thinking that there might be some kinetic energy because when it reaches the top just barely it might be moving with some tiny velocity like this. But the whole idea behind these problems is the final velocity in the final kinetic energy is going to be zero because it's kind of related to the idea of the minimum speed. So for example, let me just throw some random numbers. Imagine we had a velocity here of 20 or a minimum speed of 20 and then by the time we reached the top of the block had some speed, v. B. And that was equal to five. Well if it still has some speed here then that means that you could have actually had a smaller initial speed, this could have been less than 20 and you would have you know, you still would have made it to the top. Now, imagine I had this as 18 and then VB is equal to two. You still could have gone a little bit smaller, a little bit slower to reach the top of the hill eventually. What happens is you're gonna reach some number here and that's going to be the minimum speed. If you go any lower than any slower than that, you're actually not going to reach the top of the hill and you're gonna come crashing right back down again. So that's the whole idea is at the top of the hill, we're going to have zero kinetic energy. Now. Finally, we're also going to have some gravitational potential energy because we're at some height YB So really we're just gonna expand these last two terms here. So we have one half M. V. A final sorry, sorry, via squared equals M. G. Y. B. All right. So we can do we can cancel out these masses here. We can go ahead and solve for this via so you go ahead and re arrange for this. You're gonna get V. A. Is equal to the square root of two G times Y. B. This is really just square root of two G. Y. Final We've seen this equation before. This is super powerful. So really we've got is the minimum speed is gonna be the square root of two times 9.8 times 20. And if you go ahead and work this out in your calculator, gonna get 19.8 m per seconds. Alright? So there's no other way you could have solved this instead of uh other than using energy conservation. All right, So that's it for this one. Guys, let me know if you have any questions.