Moment of Inertia Problem

AlephZero

No. Make a drawing of a cut through the center of the sphere. The cut surface is a circle radius r. You have a little right-angled triangle, with the horizontal side length dr, the vertical side length dz, and the slant (the actual surface) is length ##\sqrt{dr^2 + dz^2}## by Pythagoras's theorem.

Of course you can use trig and get the length in terms of an angle, if you want.

Doc Al

No. See AlephZero's response. (He beat me to it. )

greswd

I get it, but, I don't understand why aleph's method doesn't apply to volume integrals.

greswd

I think I got it now.

Let's say I'm trying to find the circumference and area of a circle by using rectangles to approximate.


As the no. of rectangles approaches infinity, their combined height will always be equal to the diameter, and will never approximate the circumference.

However, the combined area of those rectangles will approximate the area of the circle.


This is why dz works for volume and not area integrals.

It's a geometric proof, but how do we prove it mathematically?

greswd

Revisiting an old thread, does the proof lie in line integrals?

sankalpmittal

Yes. Integration rocks !! As always. Also, as I can see by your thread, you are inquisitive of finding moment of inertia of a solid sphere, right ? I did not follow the hyperphysics derivation, but I derive it using moment of inertia of hollow sphere, which is 2MR²/3..

I am sorry. You ought to find moment of inertia of hollow sphere ?
There are two ways:

1. Integration (difficult)
2. Coordinate geometry (easy!).. which I think you have been told here.