00:01
Hi there, so for this problem we are told that its vertical shell has a radius, so let's call this the radius h, which is equal to 2 .84 centimeters and a cylinder with a radius, let's call this c -i -c -y, and that is 7 .47 centimeters.
00:32
Rolling without sleeping along the same floor.
00:35
The two objects have the same mass.
00:38
And so we are going to call it the mass of the spherical shell is equal to the mass of the cylinder.
00:46
And if they have the same total kinetic energy, what should the ratio of the spherical shell angular speed to the cylinder's angular speed b? so what we need to obtain is the angular speed of the shell divided by the angular speed of the cylinder.
01:10
We start by writing the total kinetic energy for the cylinder.
01:17
So we will have that for the cylinder is 1 divided by 2 times the angular momentum of the cylinder times the angular speed square.
01:28
And then this plus 1 divided by 2 times the mass times the speed of the cylinder square.
01:39
So in here we can use the definition of the moment of, yes, the moment of inertia of a cylinder.
01:58
We know that that is just simply its mass times the radio of the cylinder to the square and this divided by two.
02:08
And this times the angular speed and this plus one divided by two times the mass.
02:15
And here we can use the relation between the speed and the angular speed that we know that is just simply the product between the radius of the cylinder and the angular speed of the cylinder.
02:27
So we substitute that in here.
02:29
And of course, we need to elevate that to the square.
02:32
So if we simplify this, we will find that the kinetic energy, the total kinetic energy for the cylinder, is three times the mass times.
02:44
The radius of the cylinder square, and this times the angular speed of the cylinder square, and this divided by four.
02:59
Now, since we want to determine the angular speed of the cylinder, we just solved for it.
03:06
So let's say that this is the square of this, so that will be four times the kinetic energy divided by three times the mass of the cylinder, times the radius of the cylinder square.
03:20
Now remember that those values are given, the radius for the cylinder and the radius for the, well, let's call this equation one, and the radius for the spherical shell.
03:31
Now we pass to do the same procedure, but for the total kinetic energy of the spherical shells.
03:43
So that will be one divided by two times the moment of inertia of this spherical shell times the angular speed to the square, and this plus 1 divided by 2 times the mass, times the speed of the spherical shell to the square.
04:00
So now we just simply substitute the expression that we know for the angular momentum for the moment of the...
04:14
I'm sorry, for the moment of inertia of the.....