Spherical cap moment of inertia

In summary, the moment of inertia of a thin cap with cut circle radius r should be the same as that of a half sphere with radius r; and moreover, h has disappeared!
  • #1
André Verhecken
4
0
Hi, I'm trying to find the formula for the moment of inertia of a spherical cap relative to the axis perpendicular to its flat area. I know it should be done by (triple) integration, but it's a very long time since I've done that kind of math ! So, if someone out there knows the formula ...
 
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  • #2
André Verhecken said:
Hi, I'm trying to find the formula for the moment of inertia of a spherical cap relative to the axis perpendicular to its flat area. I know it should be done by (triple) integration, but it's a very long time since I've done that kind of math ! So, if someone out there knows the formula ...
The moment of a full sphere is 2MR^2/5. How would the moment of inertia of 1/2 a sphere be related to the moment of inertia of the full sphere?

AM
 
  • #3
That's easy: half of that, of course. But further on ? Do you suggest that the Moment Of Inertia of the cap is then to be calculated from the half sphere volume and the relation between cap volume and half sphere volume ?
AV
 
  • #4
André Verhecken said:
That's easy: half of that, of course. But further on ? Do you suggest that the Moment Of Inertia of the cap is then to be calculated from the half sphere volume and the relation between cap volume and half sphere volume ?
AV
From your initial statement of the problem, I assumed that 'cap' referred to a portion of the solid sphere.

Look at the integral for calculating the moment of inertia for a half sphere:

[tex]I = \frac{1}{2}\rho\pi \int_{0}^R (R^2 - z^2)^2 dz[/tex]

To find the moment of a portion you just integrate from z_0 rather than 0.

AM
 
  • #5
As a retired chemist, it’s a very long time (40 years) that I haven’t done any integration calculus, so I forgot quite a lot of it. But I’ll try:

Let’s consider a sphere (radius R, density ρ) from which a cap is cut with a height (= the “arrow”) h and a radius of the cut circle: r. Then I name H = R – h.

Integrating your formula for a half sphere for z = 0 to R gives the familiar I = m R2 / 5

Integration for z = 0 to H (which is the same as z = 0 to R – h) gives I = m (R – h)2 / 5

So that I (cap) = m (R2 – (R- h)2 ) / 5

= m h (2R – h) / 5

In my problem, R is unknown; only r and h are measured. But R is easily calculated as:

R = (r2 + h2) / 2 h

Substituting R in the formula for I(cap) then yields : I(cap) = m r2 / 5

This may be correct or it may be wrong (I cannot grasp it intuitively): the moment of inertia of a thin cap with cut circle radius r should be the same as that of a half sphere with radius r; and moreover, h has disappeared ! This would mean that any cap with radius of cut circle = r, cut from any sphere with radius > R, would obey this formula, and this seems impossible to me (although the formula is correct when r = R). I fear there is something fishy in my reasoning or my calculus.
I would appreciate if you would point out my error !

AV
 
  • #6
André Verhecken said:
This may be correct or it may be wrong (I cannot grasp it intuitively): the moment of inertia of a thin cap with cut circle radius r should be the same as that of a half sphere with radius r; and moreover, h has disappeared ! This would mean that any cap with radius of cut circle = r, cut from any sphere with radius > R, would obey this formula, and this seems impossible to me (although the formula is correct when r = R). I fear there is something fishy in my reasoning or my calculus.
[tex]I = \frac{1}{2}\rho\pi \int_{z_0}^R (R^2 - z^2)^2 dz[/tex]

[tex]z_0 = R-h[/tex]

Expand:

[tex]I = \frac{1}{2}\rho\pi(\int_{z_0}^R (R^4 -2R^2z^2 + z^4)dz)[/tex]

Separate the integrals:

[tex]I = \frac{1}{2}\rho\pi(\int_{z_0}^R R^4 dz -\int_{z_0}^R 2R^2z^2dz + \int_{z_0}^R z^4dz)[/tex]

Simplifying that is a bit of a chore but it should give you the result in terms of R and h ([itex]z_0 = (R-h)[/itex]). You then have to work out the volume of the cap and substitute for [itex]\rho = M/V[/itex]. It is a non-trivial exercise. This is the kind of thing that Maple was created for!

AM
 
  • #7
I did use exactly the same formulas as you gave now, but I found where I went wrong the other day: I calculated the separated integrals for R, and then presumed that the result for z° would we similar in form. So for this second calculation I forgot that R is not a variable.
I did the calculation again, and avoided the simplifying by entering the somewhat complex formula as such in Excel : it works !

Thanks for your stimulating help !
AV
 

1. How is the moment of inertia of a spherical cap calculated?

The moment of inertia of a spherical cap can be calculated using the formula I = (2/3)mR2, where m is the mass of the cap and R is the radius of the sphere.

2. What is the significance of the moment of inertia of a spherical cap?

The moment of inertia of a spherical cap is an important property that describes the rotational inertia of the cap. It is used in analyzing the motion and stability of objects that have a spherical cap shape.

3. How does the moment of inertia of a spherical cap differ from a solid sphere?

The moment of inertia of a spherical cap is smaller than that of a solid sphere with the same mass and radius. This is because a spherical cap has less mass concentrated at the center, resulting in a lower rotational inertia.

4. Can the moment of inertia of a spherical cap change?

Yes, the moment of inertia of a spherical cap can change if the mass or radius of the cap changes. It also depends on the distribution of mass within the cap. For example, a spherical cap with a thicker or thinner shell will have a different moment of inertia.

5. What is the practical application of the moment of inertia of a spherical cap?

The moment of inertia of a spherical cap is used in designing and analyzing the stability of structures and objects with spherical cap components, such as domes and spherical tanks. It is also important in understanding the physics of rotational motion and angular momentum.

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