Calculating Moment of Inertia for a Sphere: What's the Best Approach?

In summary, the conversation was about trying to calculate the moment of inertia for a sphere using calculus. The person initially made a mistake in their calculation and realized they needed to adjust for the varying radius as they integrated. They were seeking help and discussing potential approaches to the problem.
  • #1
Ja4Coltrane
225
0
This was not actually homework, but I was just trying to see if I could calculate moments of inertia and apparently, I cannot.
I'm trying to show that the moment of inertia for a sphere is (2/5)MR^2
So I started with I=(integral)(r^2)(dm)
then P=dm/dv=dm/(4pi(r^2)dr)
so dm=(4)(pi)(r^2)(P)(dr)
so I substituted into the original equation, removed constants from the integral, and substituted P for M/V=(3M/(4(pi)(R^3)))
I=12pi(M)/(4(pi)r^3)[integral]r^4 (dr)
I=(3/5)MR^2 which is wrong!

(sorry about the lack of pretty math writing)
Thanks for any help, and by the way, I'm only a high school student so my calculus knowledge is very limited (in fact, the only reason I know what integration is is because of my physics class).
 
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  • #2
Oh, I just realized something!
I am integrating as if the higher part has the same radius because it is the same distance from the center but it is closer to the axis!
now I really don't know what to do.
 
  • #3
Ja4Coltrane said:
Oh, I just realized something!
I am integrating as if the higher part has the same radius because it is the same distance from the center but it is closer to the axis!
now I really don't know what to do.
This is not a simple calculation. There are two approaches you can take with limited calculus experience. The first is to find the moment of inertia of a disk about its symmetry axis, and then slice the sphere into disks of thickness dx having a common axis that is a diameter of the sphere. Then add (integrate) the moments of inertia of all the disks. The hard part is finding the radius of each disk as a function of x, but that can be done using the equation for the surface of the sphere.

The second approach is to find the moment of inertia of a cylindrical shell about its symmetry axis (easy since all the mass has the same radius) and think of the sphere as many concentric shells of radius r and thickness dr. The hard part here is finding the length of each cylinder as a function of r, but again this can be found from the equation for the surface of the sphere.
 

Related to Calculating Moment of Inertia for a Sphere: What's the Best Approach?

1. What is moment of inertia of a sphere?

Moment of inertia of a sphere is a measure of its resistance to rotational motion. It is a property that depends on the mass and distribution of mass within the sphere.

2. How is moment of inertia calculated for a sphere?

The moment of inertia of a sphere can be calculated using the formula I = (2/5)mr^2, where m is the mass of the sphere and r is the radius.

3. What are the units of moment of inertia?

The units of moment of inertia are kg*m^2 or kg*cm^2. It is a measure of mass multiplied by the square of distance, making it a measure of mass distribution.

4. How does moment of inertia affect the motion of a sphere?

The moment of inertia determines how easily a sphere can rotate. A sphere with a larger moment of inertia will require more force to rotate, while a sphere with a smaller moment of inertia will rotate more easily.

5. Can the moment of inertia of a sphere change?

Yes, the moment of inertia of a sphere can change if its mass or distribution of mass changes. For example, if a sphere is compressed or stretched, its moment of inertia will change.

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