Work/energy problem with friction

welcome back I'll do another conservation of energy problem and this time I'll add another twist so far everything we've been doing energy was conserved by the law of conservation but that's because all of the forces that we're acting in these systems were we're conservative forces and now I'll introduce you to a problem that it has a little bit of friction and we'll see that some of the energy gets lost to friction and we can think about a little bit where where does that energy go so let's say and I'm getting this problem from the University of Oregon's is zebu org and dot edu and they seem to have some nice physics problems so I'll use theirs and I just wanna make sure they get credit so let's see they say a 90 kilogram bike and rider so the bike and rider combined or 90 kilogram so let's just say the mass is 90 kilograms 90 kilograms start at the rest start at rest from the top of a 500 meter long hill okay so I think they mean that the hill is something like this so if this is this this is the hill that the hypotenuse here is 500 meters long so the length of that this is 500 meters 500 meter long hill with a 5 degree incline so this is 5 degrees 5 degrees we can kind of just view it like a wedge like we've done in other problems there you go this pretty straight okay assuming an average friction force of 60 Newtons okay so they're just not telling us the coefficient of friction and then we have to figure out the normal force and all of that they're just telling us what is the the drag of friction or how much is actually friction acting against this rider's motion we could think a little bit about where that friction is coming from so the force of friction force of friction is equal to 60 Newtons and of course it's going to be going against his motion or her motion and the question asks us find the speed of the biker at the bottom of the hill so the biker starts up here stationary that's the biker my very artful rendition of the biker and we need to figure out the velocity at the bottom so let's figure out this to some degree is a potential energy problem but it definitely is a potential mechanical love it's definitely a conservation of mechanical energy problem so let's figure out what the energy of the system is when the when the rider starts off so the rider starts off at the top of the sail so we definitely some potential energy and is stationary so there's no kinetic energy so all of the energy is potential and what is the potential energy well its potential energy is equal to mass times the acceleration of gravity times height right well that's equal to the mass is ninety acceleration of gravity is 9.8 meters per second squared and then what's the height well here we're gonna have to break out little trigonometry we need to figure out this side of this triangle if we do insert this whole thing a triangle let's see we want to figure out the opposite we know the hypotenuse and we know this angle here so the sine of this angle is equal to opposite over hypotenuse so s oh H sine is opposite over hypotenuse so we know that the height so we let me do a little work here we know that sine five degrees is equal to the height over 500 or that the height is equal to 500 sine five degrees and I calculated the sine of five degrees ahead of time let me make sure I still have it because I didn't have my calculator with me today but you could do this on your own so this is equal to 500 and the sine of five degrees is 0.2 0.7 so when you multiply these out what do I get I'm using the calculator on Google actually 500 times sine you get forty three point six so this is equal to forty three point six so the height of the hill is forty three point six meters so going back to the potential energy we have the mass times the acceleration of gravity times the height times forty three point six and this is equal to and now I can use just my regular calculators that's an office to figure out trig functions anymore so 90 just so you can see the whole thing times 9.8 times forty three point six is equal to CE 38,400 Li thirty-eight thousand four hundred fifty-five so this is equal to 38 thousand four hundred and fifty-five joules or Newton meters and that's a lot of potential energy so what happens at the bottom of the hill and so I have to readjust my chair at the bottom of the hill all of this gets converted to kinetic question does all of it get converted to kinetic energy almost we have a force of friction here and friction you can kind of view friction as something that eats up mechanical energy or you could is these are also called non conservative forces because when you have these forces at play all of the force is not conserved so a way to think about it is is that the energy let's just call it total energy so let's say total total energy initial well let me just write initial initial energy is equal to the energy wasted in friction energy wasted in friction let me wasted I should have written just letters from friction from friction plus final energy so we know what the initial energy is in this system that's the potential energy of this bicyclist in this or you know this roughly thirty eight and a half kilojoules or thirty eight thousand five hundred joules roughly and now let's figure out the energy wasted from friction or how much essentially and the energy wasted from friction is the negative work that friction does and what does negative work mean well the bicyclist is moving five hundred meters in this direction so distance is five hundred meters but friction isn't acting along the same direction as distance friction is the whole time friction is acting again the distance so when the force is going in the opposite direction as the distance your work is negative so in this case we could say the work of friction so another way of thinking of this problem is energy initial is equal to though is equal to thee or you could say the energy initial plus the negative work of friction right if we say that this is a negative quantity that this is equal to this is equal to the final energy and here I took the friction and put on the other side because I said this is going to be a negative quantity in the system and so you should all just make sure that your if you have friction in the system just as a reality check that your final energy is less than your initial energy our initial energy is this 38 let's just say 38.5 kilojoules what is the negative work that friction is doing well it's 500 meters in the entire 500 meters it's always pushing back on the rider with a force of 60 Newtons so if force times distance so it's minus 60 Newtons because going in the opposite direction of the motion times 500 and this is going to equal the ending the this is going to equal the 500 the final energy right and what is this 60 times 500 that's that's that's 3000 no 30000 right so let's subtract 30 thousand from 38,500 so let's see minus 33 and I didn't have to do that I could do that in my head so that gives us 8455 joules is equal to the final energy and what is all the final energy well by this time the drive the riders gotten back to I guess we could call the sea level so all of the energy is now going to be kinetic energy right and what's the formula for kinetic energy it's one-half MV squared and we know what M is the mass of the rider is 90 so we have this is 90 so if we divide both sides so the one-half times 90 that's 45 so 8 4 5 5 divided by 45 so we get V squared is equal to added by 45 is equal to 180 7.9 and let's take the square root of that and we get the velocity 13.7 so the final velocity so if we take the square root of both sides of this so the final velocity is 13.7 and you can't read that thirteen point seven meters per second and this was a slightly more interesting problem because here we had a good bit of it the energy wasn't completely conserved some of the energy you could say was eaten by friction and actually that energies didn't disappear into a vacuum it was actually generated into heat and makes sense if you slid down a slide of sandpaper your pants would feel very warm by the time you got to the bottom of that but the friction of this they didn't they weren't specific on where the friction came from but it came it could have come from the gearing within the bike it could have come from the wind maybe the the bike actually skated a little bit on the way down I don't know but hopefully you found that a little bit interesting and now you can not only work with conservation of mechanical energy but you can work a work work with problems where there's a little bit of friction involved as well anyway I'll see you in the next video