Tension in an accelerating system and pie in the face

Welcome back. We just finished this problem with the pulleys and the inclined plane. And I just wanted to do one final thing on this problem just because I think it's interesting. And then we can move onto what seems like a pretty fun problem. So the last thing I want to figure out is, we figured out that this 20 kilo-- actually, the whole system will accelerate up and to the right at 4.13 meters per second squared. And then the second part of this question is, what is the tension in this rope or this wire? And at first you might say, this is complicated. You know, this thing isn't static anymore. The thing is actually accelerating. How do I do it? Well this is how you think about it. Just pick one part of the system. Let's say that all we could see was this 20 kilogram mass. So let's say all we could see was this 20 kilogram mass. And we know it's suspended from a wire. And we also know that this 20 kilogram mass is not accelerating as fast as it would if the wire wasn't there. It's accelerating only at 4.13 meters per second. If the wire wasn't there, it'd be accelerating at 9.8 meters per second, the acceleration of gravity. So the wire must be exerting some upward force on the object. And that is the force of tension. That is what's slowing-- that's what's moderating its acceleration from being 9.8 meters per second squared to being 4.13 meters per second squared. So essentially, what is the net force on this object? On just this object? Well the net force is-- and you can ignore what I said before about the net force in all the other places. But we know that the object is accelerating downwards. Well, we know it's 20 kilograms. So that's its mass. And we know that it's accelerating downwards at 4.13 meters per second squared. So the net force, 20 times-- see, times 20 is 82-- let's just say 83 Newtons. 83 Newtons down. We know that the net force is 83 Newtons down. We also know that the tension force plus the force of gravity-- and what's the force of gravity? The force of gravity is just the weight of the object. So the force of tension, which goes up, plus the weight of -- the force of gravity is equal to the net force. And the way I set this up, tension's going to be a negative number. Just because I'm saying positive numbers are downwards, so a negative number would be upwards. So tension will be what is 83 minus 196? Minus 196 is equal to minus 113 Newtons. And the only reason why I got a negative number is because I used positive numbers for downwards. So minus 113 Newtons downwards, which is the same thing as 113 Newtons upwards. And so that is the tension in the rope. And you could have done the same thing on this side of the problem, although it would have been-- well, yeah. You could have done the exact same thing on this side of the problem. You would've said, well what would it have accelerated naturally if there wasn't some force of tension on this rope going backwards? And then you're saying, oh, well, we know it would have gone in this direction at some acceleration, but instead it's going in the other direction. So you use that. You figure out the net force, and then you say the tension plus all of these forces have to equal the net force. And then you should solve for the tension. And it would be the same tension. Now we will do a fun and somewhat simple, but maybe instructive problem. So I have a pie. This is the pie. This is parallel. And I have my hand. You can tell that my destiny was really to be a great artist. This is my hand. And I'm holding a pie, and I'm looking to smash this pie into this individual's face. I actually was a, I was the newspaper cartoonist in high school, so I have some minor-- but anyway. Let's make it a bald man. Well anyway, I shouldn't be focusing on the drawing. He has a moustache. Anyway, I'm looking to throw this pie into this guy's face. And the problem is, I need to figure out how fast do I need to accelerate this pie for it to not fall down? Right? Because what's happening? Well there's the force of gravity on this pie. There's a force of gravity on this pie and if I don't accelerate it fast enough, it's just going to slide down. And I'll never be able to, It'll never reach the guy's face. So I don't want this pie to slide down at all. How fast do I have to push on it? Well, we know that the coefficient of friction-- you don't know this, but I know that the coefficient of friction between my hand and the pie, the coefficient of friction is equal to 0.8. So given that, how fast do I have to accelerate it? Well let's see what's happening. So we have the force of gravity pulling down. So let's say that the mass of the pie is m. m equals mass. So what is the force of gravity pulling down on the pie? Well the force of gravity is just equal to m times 9.8. Right? The force of gravity is equal to m times 9.8. In order for this pie to not move down, what do we know about the net forces on that pie? Well we know the net forces on that pie have to be 0. So what would be the offsetting force? Well, it would be the force of friction. So we would have a force of friction acting upwards. Right? Because the force of friction always acts opposite to the direction that the thing would move otherwise. So essentially, our force of friction has to be greater than, roughly, greater than or equal to. Because if it's greater than, it's not like the pie is going to move up. Friction by itself will never move something, it'll just keep something from being moved. But let's just figure out the minimum. I won't do the whole inequalities. The force of friction has to be equal similarly, to 9.8 times the mass of the pie. So if the coefficient of friction is 0.8, what is the force that I have to apply? Well, the force I have to apply in this case is going to be the normal force, right? That's normal to the bottom of the pie. Right? My hand is now like the surface of the ramp. So this is the normal force. And we know that the force of friction is equal to the coefficient of friction times the normal force. I'm going to switch colors because this is getting monotonous. And the force of friction, we know has to be 9.8 times the mass. So 9.8 meters per second times the mass. 9.8m is the force of friction. And that has to equal to coefficient of friction times the normal force. And remember, the normal force is essentially the force that I'm pushing the pie with. And we know this is 0.8, so we have 9.8 times the mass-- that's not meters, that's the mass-- is equal to 0.8 times the normal force. So you have the normal force is equal to 9.8 times the mass divided by 0.8. What's 9.8 divided by 0.8? 9.8 divided by 0.8 is equal to 12.25. So the normal force that I have to apply is 12.25 times the mass. So that's the force I'm applying. It's time the mass. We don't know the mass of the pie. So how fast am I accelerating the pie? Well, force is equal to mass times acceleration. This is the force, 12.25m-- that's the force-- is equal to the mass times the acceleration of the pie, right? And it's the same pie that we're dealing with the whole time, so it's still m. And you can take out m from both sides of the equation. So the acceleration, the rate at which I have to change the velocity, or the acceleration that I have to apply to the pie is 12.25 meters per second squared. And so actually, I have to apply more than 1g, right? Because g is the force of gravity. And gravity accelerates something at 9.8 seconds-- 9.8 meters per second squared. So I have to do something at 12-- I have to push and accelerate the pie at 12.25 meters per second squared. So it's something a little over 1g in order for that pie to not fall and in order for my normal force to provide a force of friction so that the pie can reach this bald man's face. I will see you in the next video.