# Moment of Inertia of Solid Sphere - Proof

• Math_Maniac
In summary, the speaker is having trouble deriving the moment of inertia of a solid sphere through its center of mass and has attached their working. They have received a different solution than what is stated in textbooks and are seeking help in identifying their mistake. The issue is traced to using two different meanings for the variable "r" and using the incorrect formula for infinitesimal volume. The speaker clarifies that this is not a homework problem.
Math_Maniac
So I have been having a bit of trouble trying to derive the moment of inertia of a solid sphere through its center of mass. Here is my working as shown in the attached file.

The problem is, I end up getting a solution of I = (3/5)MR^2, whereas, in any textbook, it says that the inertia should be equal to I = (2/5)MR^2. Is anyone able to tell me where I went wrong in my working? This is not a homework problem by the way.

Thanks.

#### Attachments

• Sphere.png
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Delta2 and T dawg
You are using ##r## with two different meanings and mixing them up. 1) The distance from the axis of rotation (the ##r## in the definition of the moment of inertia). 2) The distance from the centre of the sphere. These are not the same.

Math_Maniac said:
This is not a homework problem by the way.
Regardless of whether it is actual homework or not, it should be posted in the homework forums if it is homework-like.

Delta2
Just to add something to Orodruin's enlightening post,

In many moment of inertia calculations these two meanings happen to be the same thing (for example in the calculation of the Moment of Inertia of an infinitesimally thin circular disc, the distance from the axis of rotation (that passes through the center and is perpendicular to the plane of the disc) equals the distance from the center of the disc ) BUT in the general case they are not the same thing and in this case they are not the same thing.

What is the equation that relates r' (the distance from the axis of rotation) and r (the distance from the center of the sphere) in this case?

Also the dV you calculate is not the same as the dV that appears in the definition of the moment of inertia. You calculate the infinitesimal volume between a sphere with radius r and radius r+dr. But the dV in the integral in the definition of the moment of inertia is ##dV=r^2\sin\theta dr d\theta d\phi## (r is the distance from the center of the sphere). You just can't use your definition of dV because if you find the equation of r' correctly you ll see that it depends on ##\theta## and ##r## and not only ##r##.

Last edited:

## What is the moment of inertia of a solid sphere?

The moment of inertia of a solid sphere is the measure of its resistance to changes in rotational motion. It is a property that depends on the mass distribution and geometry of the sphere.

## How is the moment of inertia of a solid sphere calculated?

The moment of inertia of a solid sphere can be calculated using the formula I = (2/5) * M * R^2, where M is the mass of the sphere and R is the radius of the sphere.

## What is the significance of the moment of inertia of a solid sphere?

The moment of inertia of a solid sphere is an important concept in physics and engineering, as it helps in understanding the rotational motion of objects and predicting their behavior. It is also used in designing structures and machines that involve rotational motion.

## How does the moment of inertia of a solid sphere compare to other shapes?

The moment of inertia of a solid sphere is higher than that of other shapes with the same mass, such as a hollow sphere or a disk. This is because the mass is distributed farther from the axis of rotation in a solid sphere, making it more resistant to changes in rotational motion.

## Can the moment of inertia of a solid sphere change?

Yes, the moment of inertia of a solid sphere can change if there is a change in its mass distribution or geometry. For example, if the sphere is deformed or has a non-uniform mass distribution, its moment of inertia will also change.

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