Calculating rotational inertia of a sphere

In summary, calculating the rotational inertia of a sphere involves dividing the sphere into thin disks and integrating the moment of inertia of each disk, which is determined by its mass and distance from the axis of rotation. The integration can be simplified by using spherical coordinates and replacing the density with the mass over volume ratio of the sphere.
  • #1
Silimay
Just how do you calculate the rotational inertia of a sphere?
Assuming the sphere lies at the center of the xyz coordinate system, I divided the sphere into a series of cross-sections of verticle width dz and area pi*y^2. I then multiplied these together and multiplied this by z^2, and multiplied this by density (M/V, or M/(4/3*pi*R^3)), and then tried to integrate with respect to z from -R to R. I wasn't sure whether or not to include z itself in the integration (z=(R^2-Y^2)^(1/2)). I have a feeling I completely messed the entire problem up; however, I'm not sure where. Did I go about doing it in an entirely wrong way? Should I use double integrals (would that be easier)? Do I have to use spherical coordinates or something?
Any help is appreciated. :smile:
 
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  • #2
Your answer lies here

Hopefully Arildno and Krabs' approach are within your level of understanding...
 
  • #3
Thank you thank you thank you! That was REALLY helpful...I've been trying to understand the process of calculating inertia for days...my textbook was of no help. I think I finally understand it.
 
  • #4
Silimay said:
Just how do you calculate the rotational inertia of a sphere?
Divide the solid sphere into thin disks of thickness dz and mass dm. For the thin disk is [itex]I = \frac{1}{2}MR^2[/itex]

The moment of inertia of each disk is
[tex]dI = \frac{1}{2}x^2dm[/tex] where [tex]dm = \rho \pi x^2 dz[/tex]

So [tex]dI = \frac{1}{2}\rho \pi x^4 dz[/tex]

Then integrate dI from z = -R to R (note: [itex]x^2 = R^2 - z^2[/itex])

That will give you I in terms of [itex]\rho[/itex] which is M/V (where V is the volume of the sphere and M is its mass) so just replace [itex]\rho[/itex] with M/V.

The integration looks a little tough because of the [itex](R^2 - z^2)^2[/itex] Good luck.

AM
 

FAQ: Calculating rotational inertia of a sphere

What is rotational inertia?

Rotational inertia, also known as moment of inertia, is a measure of an object's resistance to rotational motion. It is determined by the object's mass and the distribution of that mass around its axis of rotation.

How do you calculate rotational inertia of a sphere?

The formula for calculating the rotational inertia of a solid sphere is I = (2/5) * m * r^2, where I is the moment of inertia, m is the mass of the sphere, and r is the radius of the sphere.

What is the unit of measurement for rotational inertia?

The unit for rotational inertia is kilogram-meter squared (kg*m^2) in the SI system of units. In the imperial system, it is pound-square feet (lb*ft^2).

How does the distribution of mass affect the rotational inertia of a sphere?

The farther the mass is from the axis of rotation, the higher the rotational inertia of the sphere will be. This means that a sphere with a more concentrated mass near its center will have a lower rotational inertia compared to a sphere with the same mass but spread out towards the edges.

What is the significance of calculating rotational inertia of a sphere?

Knowing the rotational inertia of a sphere is important in understanding how it will behave when subjected to rotational motion. It also plays a crucial role in various applications such as designing vehicles, machines, and sports equipment.

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