Moment of Inertia Problem

greswd

Here's a derviation from HyperPhysics:



He says:

[tex]dV=πy^{2}dz[/tex]


However, if we're finding the surface area of the sphere:

[tex]dA=2πRdz≠2πydz[/tex]


If we cannot use [tex]dA=2πydz[/tex], how come [tex]dV=πy^{2}dz[/tex] is still applicable?

Doc Al

That's the volume of those circular disks.

What does finding the surface area of the sphere have to do with this?

greswd

If I want to find the MoI of a hollow sphere.

Based on the same assumptions [tex]dA=2πydz[/tex], it should be like that, but it's not .


I'm trying to calculate the MoIs of some hollow objects, not sure what kind of area formula I should use.

Doc Al

Not sure what you mean by "the same assumptions". To find the surface area of that ring, you need the arc length, not the vertical distance (dz).

greswd

Yup, that's the perplexing part of the formula [tex]A=2πRz[/tex]



As z tends to zero, that particular segment of a sphere should tend toward a segment of a cone. And for a cone, when z tends to zero

[tex]A→2πrz[/tex]

The difference between the conical area and the area of a cylinder wrapped around it should decrease until they are equal when z tends to zero.

Doc Al

Again, it's the length of that side that must be used. And that length is not simply z. (It's at an angle.)

Their ratio remains finite. They aren't the same.

HomogenousCow

Is this for the moment of inertia of a sphere?

Doc Al

A spherical shell.

HomogenousCow

I'm pretty sure it's for a full sphere, that dI there represents the moment of inertia of a single disk at height z, and then he integrates from the bottom to the top.

Doc Al

Yes, that would be correct if he were finding the moment of inertia of a solid sphere, but he's not.

greswd

That sounds correct.

But why does this not apply to volume integrals?

Doc Al

Any slant with the side will be a second order correction to the radius and will have no effect on the volume element.

The volume is the area times thickness:
dV = area * dz = πr²dz

If we vary the radius by dr, we get:
dV = π(r + dr)²dz = π(r² + 2rdr + dr²)dz
dV = πr²dz + 2πrdrdz + πdr²dz

The higher order differentials can be ignored.

greswd

So ignoring the higher order differentials, we still get an exact answer?



What if the external area is perimeter times thickness:

dA = perimeter * dz =2πrdz

Varying the radius by dr, we get:

dA = 2π(r+dr)dz =(2πr+2πdr)dz = 2πrdz + 2πdrdz

Doc Al

Yes, that's how it's done. In the limit as they go to zero, (dx)² can be ignored compared to dx.
But the surface area is not perimeter times thickness; it's perimeter times the length of the side.

greswd

ah, that makes sense.

but then volume can also be considered area times some funny ratio and not just the vertical height dz

Doc Al

You can express the length of the side in terms of the height. But that will also depend on the angle.

greswd

yeah it will. so you've already explained why we can use dz for volume integrals, by ignoring the higher order differentials.

but for external area, you said we must account for the slant, I used (r + dr), doesn't that do the job?

I'm afriad my mathematical understanding is really sketchy.