Conservation of energy (part 2)

0:00- [Instructor] Suppose we drop a block of mass M from some height H. What's gonna happen? Well, let's say that the ramp is frictionless. There's no force of friction over here. But once you come to this horizontal surface, that is a rough surface. There is friction, and the coefficient of sliding friction is mu K.

0:17So what do you think is gonna happen? Well, you can imagine the block will speed up over here, and because of friction then, it will slow down and eventually come to a stop. So it's gonna look somewhat like this, right? So the question we wanna try and answer is how far does it get displaced before coming to a stop?

0:33Whenever you have problems like this, which does not involve time, you can see we don't have to worry about time over here, we can, a cool way to solve such problems is by using the idea of energy. To be very specific, by thinking about mechanical energy. Work done on a system changes its mechanical energy, and that would be a starting point to solve this problem.

0:54So let's do this. So the first step is to choose a system. Well, we are free to choose whatever system you want, but let's make sure that our system involves a block. So let's choose our system to be the block-earth-surface. This way what we're doing is all the forces involved, like the force of gravity or the force of friction over here, they're all going to be the part of our system.

1:19Whenever forces are part of your system it tends to make things slightly easier. All right, now that we have our system, next we need to think about its mechanical energy. What is the mechanical energy of this system comprised of? Well, mechanical energy is the kinetic energy plus potential energy.

1:34So it involves the kinetic energy of all the particles. So we have kinetic energy of the block, plus kinetic energy of the earth. Well, since earth is not moving, we can ignore that, and kinetic energy of the earth's surface. Again, we can ignore that. So it'll involve the kinetic energy of the block.

1:48And then we would think about the various potential energies. Well, which is the conservative force acting over here? Gravity is the conservative force. So we have gravitational potential energy. Any other conservative force part of our system? No, so our mechanical energy is gonna be kinetic energy plus the gravitational potential energy.

2:09Okay, now, if there are any non-conservative forces or external forces doing work on my system, it will change its mechanical energy. This means the next step is to figure out which forces are doing work on my system. Well, the forces that are involved over here, well, is the force of gravity.

2:27But I don't have to worry about gravity because gravity is a conservative force, and it's a part of my system. It conserves mechanical energy. So it doesn't change it. I've already accounted for the work done by gravity, by including potential energy over here.

2:40So I don't have to worry about gravity. It's a conservative force, which is part of my system. Are there any other forces? Yes, there's a normal force, but normal force does not do any work because normal force is always perpendicular to the direction of the motion.

2:53So the work done by normal force is zero, so I don't have to worry about normal force. Oh, then there is a force of friction over here. The force of friction will always oppose sliding over here. So the sliding friction will be to the left. But wait, friction is also a part of our system, right?

3:08Yeah, but friction is a non-conservative force. It does not conserve mechanical energy. So whenever you have non-conservative forces acting, whether it's a part of your system or not, it does not conserve mechanical energy. In fact, some of the mechanical energy actually gets dissipated as thermal energy.

3:27So friction is definitely doing work on my system, changing its mechanical energy. Is there any other force that's doing work on my system? No, that's it. So the only force that's doing work on my system, changing its mechanical energy is the frictional force.

3:44So we can now write that change in mechanical energy, Delta E, equals the work done by friction. All right, so what is changing mechanical energy? Well, it's going to be final mechanical energy minus initial mechanical energy. And that equals the work done by friction.

4:00What's final mechanical energy? Well, it's the final kinetic energy plus the final gravitational potential energy minus initial kinetic energy, plus the initial gravitational potential energy, and that equals the work done by the friction. Okay, but what is final and what's initial over here?

4:16Again, that's completely up to us. Since the block starts from here and then ends over here, well, let's just call this as our initial configuration and, or our initial state, and we can call this as our final state. Okay, so now the picture is very clear. If you would be a great idea to pause the video and see if you can substitute and you know simplify.

4:39All right, let's do this. So what's the final kinetic energy? Well, that's the kinetic energy at this point. At this point the block has come to a stop. So the final kinetic energy of the block is zero. So our system's final kinetic energy is zero. What's the final gravitational potential energy of the system?

4:57Well, when the block is on the ground, we can choose that as our reference for zero potential energy whenever we are very close to earth. That's a great reference for your zero of gravitational potential energy. So our block is on the ground over here, so we can say gravitational potential energy is also zero.

5:15So the final mechanical energy became zero. What's the initial kinetic energy? We are dropping the block from rest. So the initial kinetic energy is also zero. What is the final, sorry, what is the initial gravitational potential energy over here? Ah, when the block is at some height H from the ground, and we are very close to the earth, the gravitational potential energy is just MGH where M is the mass of the block, and G is the acceleration due to gravity.

5:48And so that's my left hand side. That should equal the work done by the force of friction. Well, that's just going to be the force of friction times the displacement, and there's gonna be a negative sign because they're in the opposite direction. So it's gonna be minus force of friction times the displacement D.

6:04And look, we have D in our equation, so we have to solve for D. So let's simplify. The left hand side just simplifies to negative MGH. And what happens to the right hand side? Well, we have this force of friction. Do we know the expression for force of friction?

6:21Yeah, it's mu K times the normal force. So it's gonna be mu K times, what's the normal force over here? Well, we have to consider the normal force over here because this is where the friction acts. So over here, normal force is just going to be equal to the gravitational force because there's no acceleration in the vertical.

6:39The forces are balanced in the vertical. So gravitational force is mg. Normal force is also going to be equal to mg. So it's going to be mu K times normal force, which is MG. So this is the force of friction times D. And MG cancels out, and there you have it, the negative sign cancels out.

6:58And so D equals H divided by mu K, and we have found the expression for this distance. Okay, let's do a sense check. That's always a great idea to do a sense check over here. We find that D is directly related to height, more height, more the distance. Does that make sense?

7:18Yeah, I mean if the block started higher by the time it comes here, it would have much higher speed, and so we would expect it to travel much farther before coming to a stop. If the block started lower, it would have much lower speed, and so you would expect it to come to a stop much closer.

7:35So it makes sense that D is directly related to height. Okay, we find that D is inversely related to the coefficient of kinetic friction. Does that make sense? Yeah, if this number is higher, the friction force is higher, and it'll stop sooner. So more the value of mu K, smaller the value of T, smaller the value of mu K, smaller the friction, and it will, you know, it'll travel much farther before coming to a stop.

8:01So it makes sense. Also, we can take a look at some edge cases. What if H equals zero? Then D goes to zero. Does that make sense? Yeah, if the H is zero, that basically means we are just keeping the block over here, and so it wouldn't travel at all obviously. So the D goes to zero, so that makes sense.

8:21If mu K is zero, we get a zero in the denominator and D kind of becomes what? Infinity. What does that even mean? Well, if mu K is zero then there is no frictional force, and so the block would then continue to move forever. That's what it's saying. That makes perfect sense.

8:39Finally, what if mu K was infinitely large? Well, then D would be almost zero. Does that make sense? Yeah, if mu K was like infinitely large, then you would have like super, super strong force of friction, and the block would almost instantly decelerate come to a stop, making D zero.

8:55So it all makes sense. We've looked at it from multiple angles. This makes a lot of sense. Okay, the final thing we can do is we can choose a different system altogether and then see if we get the same answer. I mean we should get the same answer, right? That's a good way to check our understanding.

9:11So what if we just considered the block as our system? Okay, then how would things change? Well, we can try to do this in our heads. First of all, if we only consider the block as our system, then there's only one particle in our system, then the mechanical energy would just be the kinetic energy.

9:28Okay, so there will be no concept of gravitational potential energy 'cause gravity is no longer part of our system, okay? So let's look at what happens to the left hand side of the equation. The change in mechanical energy would just be the final kinetic energy minus the initial kinetic energy, and they're both zero.

9:44So left hand side would just be zero. But what would the right hand side be? Well, the right hand side is the work done on the system. Friction is of course doing work done just like before on our system. But this time gravity will also do work on our system because gravity is no longer a part of our system.

10:01But what is the work done by gravity? Well, work done by gravity is the force of gravity times the vertical displacement, which is MGH. So notice this time on the right hand side, I will not only have this term, but I also have a plus MGH, which will eventually give me the same equation, right?

10:23And so I would eventually get the same result. So it's always a great idea when you're practicing to choose multiple systems and then just work it out and just to see that you get the same answer.