Moment of Inertia for a Ball Shell: Calculate & Show w/ Integral

In summary, the moment of inertia for a ball shell can be calculated using the integral formula I= \int_S r^2 dm, where r is the distance from the axis of rotation, R is the radius of the sphere, and \sigma is the surface mass density. By using spherical coordinates and the area element da = R^2\sin \theta d\theta d\phi, the final answer is 2/3mR^2. A picture can be used to better understand the calculation.
  • #1
mick_1
5
0
moment of inertia for a ball shell

Is ther anyone that can explain how to calculate moment of inertia for a ball shell?? I know that the answer is 2/3mR^2, but how can it be showen with integral??
 
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  • #2
Draw a picture and use spherical coordinates. Set the z-axis as the axis of rotation.

[tex]I=\int_S r^2 dm = \int_S (R\sin \theta)^2 \sigma da[/tex]

where R is the radius of the sphere and [itex]\sigma[/itex] the surface mass density.

Note that [itex]da = R^2\sin \theta d\theta d\phi[/itex]:

[tex]I= \sigma R^4 \int_0^{2\pi}\int_0^\pi \sin^3 \theta d\theta d\phi=\frac{8}{3}\pi \sigma R^4=\frac{2}{3}MR^2[/tex]
 
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  • #3
What do you mean by [itex]dV = R^2\sin \theta d\theta d\phi[/itex] ??
 
  • #4
It's an element of volume in spherical coordinates.
 
  • #5
Yeah, although it should be AREA element: [itex]da[/itex], I'll change it now...
 
  • #6
I don't understand this calculation with [itex]da = R^2\sin \theta d\theta d\phi[/itex]
??
Do you have a picture to show this??
 
  • #7
mick_1 said:
Do you have a picture to show this??

http://www.usd.edu/phys/courses/phys431/notes/sphercoor.gif one I found on a google search. The area element is just the top face of the shaded box.
 
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  • #8
http://mathworld.wolfram.com/SphericalCoordinates.html

It is shown at eqn 14. Note that the notation for the angle of declination (zenith) and the azimuth are interchanged w.r.t. mine ([itex]\phi \leftrightarrow \theta[/itex])

EDIT: Ah! Spacetiger has exactly the kind of picture I was looking for.
 

Related to Moment of Inertia for a Ball Shell: Calculate & Show w/ Integral

1. What is Moment of Inertia for a Ball Shell?

The Moment of Inertia for a Ball Shell is a measure of an object's resistance to changes in its rotational motion. It is calculated using the object's mass, shape, and distribution of mass.

2. Why is it important to calculate Moment of Inertia for a Ball Shell?

Calculating the Moment of Inertia for a Ball Shell is important because it allows us to understand how the object will behave when it is rotating. It can also help in determining the object's stability and how much force is needed to change its rotational motion.

3. How do you calculate Moment of Inertia for a Ball Shell?

The Moment of Inertia for a Ball Shell can be calculated using the formula I = 2/3 * mr², where I is the moment of inertia, m is the mass of the object, and r is the radius of the spherical shell.

4. Can you show the calculation of Moment of Inertia for a Ball Shell using an integral?

Yes, the integral for calculating the Moment of Inertia for a Ball Shell is ∫ r²dm, where r is the distance from the axis of rotation and dm is the differential mass element. This integral is then evaluated over the entire object to obtain the total Moment of Inertia.

5. How is Moment of Inertia for a Ball Shell different from other shapes?

The Moment of Inertia for a Ball Shell is different from other shapes because it is a hollow spherical shell, which means its mass is distributed evenly at a distance from the axis of rotation. This results in a simpler calculation compared to other shapes with more complex mass distributions.

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