00:01
Same radius of earth, but it has a variable density of the planet.
00:05
So here's my representation of that.
00:08
It's densest at the middle, and then it gets less sense as we move away from the center of the planet.
00:13
And then we're asked to find what the acceleration due to gravity would be on the surface.
00:18
So the variable densely doesn't actually really matter for that, because we're looking outside of the center, outside of the planet itself.
00:27
And because the density distribution is symmetric, we still know that the center of mass will be right at the middle of the planet.
00:35
So we can just reduce the situation to a point particle with the mass of the planet.
00:42
And then a distance are, maybe i'll just call the radius r.
00:47
A distance r away, we want to know what the acceleration is at that point.
00:52
So no difference.
00:53
And so, as we've seen many times, we know that this acceleration is just going to be the gravitational constant times the mass of this point and then divided by the distance squared.
01:07
So we know the radius, it's the same as earth, so we just need to find what this mass is.
01:13
And so that's where the density distribution will come into play, but beyond that it doesn't matter.
01:18
If they had told us what this mass is, it doesn't matter at all that the density is variable.
01:23
So how are we going to find this mass? so we can say that this mass is just going to be the density of the planet times the volume.
01:37
But because the density is variable, that means we'll need to integrate some.
01:41
We'll need to write a differential volume term and then integrate over that full volume to find what the masses.
01:49
So we want to set this up like an integral of the density times a differential volume element.
01:55
And then we're integrating over the full volume of the planet, and that'll give us the total mass.
02:02
But fortunately, this density distribution is symmetric, so it's spherically symmetric and varies only linearly with r.
02:11
And so that means we can actually set this up then as integrating spherical shells.
02:17
So we're kind of drawing little spheres in here, and then we have r go outwards.
02:25
So then we integrate these out to get the volume.
02:28
And so that'll make this function a little bit easier.
02:31
So rather than integrating over a full volume, we just integrate over the radius of the sphere.
02:37
And so the surface area of one of these shells is just 4 pi r squared.
02:46
So just the surface area of a sphere.
02:48
We want to use little r here since this is a variable.
02:51
We're doing an infinite number of these little shells.
02:54
And then we still have our density, and we integrate with respect to r.
02:59
So this integral looks a little bit more doable.
03:02
We just now need to find what this density row is, then we can plug that in.
03:09
So we're given the density at the center of the planet, so i'll call that row not, and then we're also given the density at the surface, so row sub -s.
03:20
And so we want to find then what row is a function of r.
03:24
Is? well, we're told that it's a linear density function, it increases linearly, so presumably then it takes the form of ax plus b, and i'll say a .r, since r is the variable we're using here.
03:39
So we know this is going to look like a -r plus b.
03:42
So we just need to find what a and b are, and then we'll have our density function...