Calculating Moment of Inertia for a Thin-Walled Spherical Object

[Karol]

Homework Statement

How to calculat the moment of inertia of a thin wall, the surface area of a ball of mass m and radii a.

Homework Equations

I=mr²

The Attempt at a Solution

Spherical coordinates.

$$dm=\frac{m}{4\pi a^2}\cdot a^2\cdot d\theta d\phi$$

See drawing.

$$I=\int_{\theta=0}^{2\pi} \int_{\phi=-\frac{\pi}{2}}^\frac{\pi}{2} dm\cdot a^2$$
$$I=\int_{\theta=0}^{2\pi} \int_{\phi=-\frac{\pi}{2}}^\frac{\pi}{2} \frac{m}{4\pi a^2}\cdot a^2\cdot a^2 d\theta d\phi$$
$$I=\frac{a^2m}{4\pi}\int_{\theta=0}^{2\pi} \int_{\phi=-\frac{\pi}{2}}^\frac{\pi}{2} d\theta d\phi$$
$$I=\frac{2\pi m a^2}{4}$$

The answer should be:

$$I=\frac{2ma^2}{3}$$

 

[tiny-tim]

Hi Karol!

You need to calculate https://www.physicsforums.com/library.php?do=view_item&itemid=31" about an axis, not about the centre

(also, the surface element is sinφdθdφ, not dθdφ)

 

[Karol]

I understand about the axis, but Why is the surface element sinφdθdφ, not dθdφ?

 

[tiny-tim]

Because the surface element from θ to θ+dθ and from φ to φ+dφ is a tiny rectangle whose sides are rdθ and rsinφdφ …

making an area of r²sinφdθdφ

(btw, mathematicians usually define θ and φ the other way round, so you'll see usually see sinθdθdφ instead )

 

[Karol]

I understand the rdθ, but why the sinφ in rsinφdφ?

 

[tiny-tim]

I understand the rdθ, but why the sinφ in rsinφdφ?

oops! I got the sinφ in the wrong place … it should be rdφ and rsinφdθ.

(Because, at latitude φ, the circle of latitude has radius rsinφ, so a slight change from θ to θ+dθ only takes you a distance radius time angle = rsinφdθ …

so that tiny rectangle has sides rdφ and rsinφdθ)

 

[Karol]

I understand, but then it should be rcosφdθ!

 

[tiny-tim]

We usually measure φ from the pole, ie 0 ≤ φ ≤ π, and then it's rsinφdθ … see http://en.wikipedia.org/wiki/Spherical_coordinates"

Only if you measure φ from the equator, ie -π/2 ≤ φ ≤ π/2 (not recommended), is it rcosφdθ.

 

[Karol]

Thanks a lot, you are a charming man (or woman...)

 

[tiny-tim]

no, fish!