00:01
The base radius of the homogeneous cone is given as base radius is given as capital r and the height, height is given as capital h and we have the total mass, total mass of the cone, right circular cone is small m.
00:29
So, first of all let us draw the diagram.
00:32
So, this is the diagram and we will draw and we will find the moment of inertia with the help of this.
00:38
So, let us say we first of all calculate the density which is rho and it is equals to mass over volume and volume of this cone will be 1 by 3 pi radius square into height.
00:55
So, density will be equals to 3m over pi capital r square h.
01:04
Let us say this is equation.
01:08
Now we can also calculate density with the help of that small strip we have considered.
01:17
So, it will be dm, the small mass of that strip divided by the small volume dv.
01:24
So, from here dm will be equals to rho into dv and dv will be equals to pi y square into dx, the volume of the small cylindrical strip.
01:45
Now we can calculate the moment of inertia with respect to the axis of this small cylindrical strip.
01:52
It will be 1 by 2 into the mass dm into the distance from the axis which is y square.
02:01
This is the moment of inertia of a thin disc.
02:04
So, we have the value of dm.
02:07
So, let us substitute that 1 by 2 into rho pi y square into dx into y square.
02:18
So, from here we can calculate the value of i, the moment of inertia i will be equals to 1 by 2 rho pi limit going from x equals to 0 to x equals to capital h and in the inside we will have y to the power 4 dx.
02:41
Now we can find out the relation between y and x...