00:01
Okay, three train cars are coupled together, and they have the masses given, 30 ,000, 50 ,000, and 20 ,000 kilograms.
00:11
A force is applied to the locomotive of 127 ,000 newtons.
00:17
And that is applied at car a.
00:22
And then we have some friction between the wheels and the rails of a coefficient of 0 .1.
00:32
So we've got car c, car b, car a.
00:45
They're all connected, and there's a force pulling car a, right? and there's friction between the wheels and the rails.
00:59
So i want to draw a free body diagram for each of the cars first off.
01:05
So i'm just going to draw the diagrams just beneath each car.
01:10
So for car c, we have the normal force.
01:16
Call n sub c.
01:18
We have the gravitational force, mass times gravity.
01:22
We have a frictional force, f sub c, because it says there's kinetic friction between the wheels and the rails.
01:35
And since the car is being pulled, we know the friction isn't going to be helping the car.
01:42
Move forward, right? it's not a, if it had an engine and the wheels were spinning, then the friction would be helping the car move forward.
01:49
But since the cars are just being pulled, that tells us that the friction is going to be fighting the change and not helping move the cars.
02:00
So we'll have some friction pulling it back, and then we have the, like, tension force between car c and car b.
02:11
I'll just call that f sub c b.
02:16
So it's the force between b and c.
02:19
Okay, for carb, i'm going to have to shift everything just a little bit here.
02:24
Same idea.
02:27
We have the normal force, gravity.
02:40
We have friction.
02:51
I'm kind of running out of room here.
02:53
Let's see.
02:57
I think i'm going to have to just move this all over a little bit.
03:00
Okay, let's try again.
03:02
So this is for car b.
03:04
I was going to do them right underneath, but i guess that's not going to work.
03:11
Okay.
03:15
So to the left, we're going to have the frictional force pulling back on car b.
03:21
And we're also going to have the force from car b, or from car c, pulling back, right? so if you think about this rope or rod, whatever it is, in this nice idealized situation, the tension there is going to be the same, right? so it's going to be pulling car b or car c forward, but it's going to be pulling car b backward.
03:46
So these forces, which i've called them the same notation, f sub c, b, they are the same, but in opposite directions.
03:55
So they're newton's third law pair.
03:57
And then pulling it forward, we have the force of, we'll call it, it's um f sub b a okay for the a car we have normal force gravity similar idea here is for car b or the friction plus the force between a and b and then pulling that we just have the applied force so i'll just call that f okay there's our free body diagrams um we want to find the accelerators of car b.
04:41
So there's two ways to do this.
04:42
The first it would be to realize that this is a coupled system.
04:47
So we could treat this as our system, that big block of cars, and then say some of the forces equals mass time acceleration, because they're all accelerating together, right? so we could do that.
05:14
And that's a really easy way to do it.
05:18
Or we could look at each, we could look at car b by itself, and then we'd have to solve a few different free body diagrams.
05:26
So that's the way we're going to do it, because that will give us the answers for c and d as well.
05:32
So we want the acceleration of car b.
05:35
So the sum of the forces on car b is equal to mass of car b times its acceleration, right? okay, what forces do we have on car b? well, to the right we have the force of b, or the force between b and a, minus the frictional force, minus the force between c and b, now equal m sub b times acceleration...