00:01
Now in this question, we are looking at a disk, right? so let me try to draw this disk first.
00:06
So this is something like this.
00:08
Okay, so this is our disk.
00:11
And the scent of this disk is here, and i'm going to use a different color to say it.
00:17
So this is x and suppose this is actually y.
00:23
And this disc has a radius, which is actually two, right? so radius of this actually is two, right? and we are basically looking at this quadrant, and there are some lamina on this quadrant, right? and the laminar has a density in this quadrant that, let me write as a row, the density, which is a function of x -y, which is given by 3x squared plus y -square, right? basically, you also found the total mass of this lamina, right? well, the total mass, let me write it as m, right? and, of course, this m is going to be given by an integration of d.
01:00
X and d y right in the first quadrant right the integration should be in the first quadrant and then row times yeah that would be it right row basically integration over row right and of course it's bad to do this using the polar coordinates right so that will be given by d theta this is the angle polar angle is from this to this right from zero to half pi right and then you of course have also the r0, right, d r times r from 0 to the radius, which is 2, right? and then times the row, row can be written at 3 times r2, basically, right? so finally, we find this to be given by the integration of a theta is given you, give you just a half pi, right? and the integration from arrow is given written as r to the cubic, right? to cubic.
01:54
And of course, this gives you 3 pi over 2 and times this integration can be the dam that gives you actually four, an all to the fourth, right? so, no, this is two, right? sorry, it's a two.
02:06
So two to the fourth rate.
02:08
So, of course, that gives you actually four here, right? and so you'll find that in the end to be given by six pi, right? so the answer actually is six pi, right? so that's the total mass.
02:24
And how about the center of mass? well, to find the center of mass, we can do the same thing, right? so the center of mass, for sure, it's going to, by symmetry, you can see that the center of mass must be somewhere along this symmetry line, right? it must be somewhere here, right? it's for sure it must be somewhere here on the, on the symmetry line.
02:50
And we can find this, of course...