00:01
All right, so in order to solve this problem, we need to look at two separate cases.
00:06
So what we're looking for is the value of p, the maximum value of p, before any sliding happens.
00:13
And there are two cases where this thing can slide.
00:16
The first case is where blocks b and c move together.
00:27
And in this case, the friction between b and c, because the only force acting on c, or rather forces acting on c, are going to be the friction between b and c and the weight of c down the ramp.
00:47
So if b and c move together, that means that the downward weight of c plus the friction force on c from b will be greater than the friction force between c and the ramp.
01:03
So in that case, they'll move together.
01:07
And in case two, block b will move out from between a and c.
01:16
So b moves, but the other ones are staying put.
01:21
And so in this case, this happens when the friction between b and c plus the weight of c down the ramp is less than the friction force that is between c and the ramp.
01:37
So first of all, let's figure out what the three different friction forces are.
01:43
There's going to be friction between blocks a and b.
01:46
That's going to be equal to the coefficient of friction between a and b times the weight, the perpendicular part of the weight force from a, because that's going to be the normal force between a and b.
02:02
So the weight force of a is the mass of a times gravity times the cosine of the angle.
02:10
And that is going to be equal to the normal force between a and b.
02:15
And so we just need to plug in our values here.
02:17
So this is going to be equal to 0 .3 times 30 kilograms.
02:22
And i'm using 9 .8 for my value for g.
02:27
If you use a different value, like 10 or 9 .81, if you want to be more precise, then just make adjustments as you need.
02:36
And if we plug those values in, we get a coefficient of friction of 76 .4 newtons.
02:47
So that's the friction between a and b.
02:50
Now, the friction between b and c is going to be slightly more complicated, because we're going to have both the weight of a and the weight of b pushing down.
03:00
So the normal force between b and c is actually the weight of both a and b.
03:05
So we can set up a very similar equation, the coefficient of friction between b and c times the combined mass of a and b times gravity times the cosine of the angle.
03:21
And so we just need to plug in our values again.
03:23
So 0 .4 this time, because that's the coefficient of friction between b and c.
03:28
And then we have 30 kilograms from a and 50 kilograms from b times 9 .8 times the cosine of 30 degrees.
03:40
And for that one, we get 271 .6, 271 .6 newtons.
03:48
And then finally, let's look at the coefficient of friction between c and the ramp.
03:58
So that's going to be the coefficient of friction, or the friction force between c and the ramp.
04:03
So that's coefficient of friction between c and the ramp, which is 0 .45.
04:06
And then this time, we have all three masses.
04:08
So ma plus mb plus mc times g times cosine of the angle.
04:18
So we have 0 .45 times 30 plus 50 plus 40 times 9 .8 times cosine of 30 degrees.
04:31
And we get a value for that of 458 .3, 458 .3 newtons.
04:42
So let's start with case b.
04:46
So let's look at case b.
04:48
So for that one, we need to know the weight force down the ramp of c as well.
04:53
So let's look at that.
04:55
And the weight force, so let's call that fg of c, is going to be the mass of c times g times the sine of 30 degrees, sine of that angle.
05:08
So that's going to be 40 times 9 .8 times the sine of 30, which is 1 .5.
05:16
And we get a weight force down the ramp of 196 newtons.
05:23
So if we look at case one, remember that case one, the friction force and the weight force of c combined have to be more than the friction force between c.
05:39
So that means we're going to be adding this value and this value...