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

• Ja4Coltrane
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.
Ja4Coltrane
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).

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.

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.

## 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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