Do Solid Cylinders and Cylindrical Sections Have the Same Rotational Inertia?

[Zarathustra1]

Let's assume for a moment that we have two objects: a solid cylinder rotating about the center, and a section of a cylinder with a given angle rotating about the center of the would-be cylinder (had it been a full cylinder). They have an equal radii, and we will modify their mass-densities in such a way that the cylinder and the cylindrical section have equal masses.

We can conclude that the mass distribution across the radius is equal in both objects. As such, would it be safe to assume they have the same rotational inertias?

If I did my math correctly, then yes. I attempted to derive a formula for the rotational inertia of a cylindrical section, and it turned out to be same formula for that of a full cylinder: .5MR^2. I was a little intrigued to see that the angle measure canceled out in the process, but it sort of makes sense. Anyone want to verify this?

(Although this question occurred to me after doing a related homework problem, it isn't a homework problem itself, so I hope this is the proper forum for this question.)

 

[binarybob0001]

Is this section of a cylinder rotating about its center of mass? Is the section shaped sort of like a piece of pie?

 

[charng]

I can confirm that your conclusion is correct. The rotational inertia of an object is directly proportional to its mass and the square of its radius. Since both the solid cylinder and the cylindrical section have the same mass and radius, they will also have the same rotational inertia. This is because the mass distribution across the radius is what determines the rotational inertia, not the shape or angle of the object.

Your derivation of the formula for the rotational inertia of a cylindrical section is also correct. This formula is derived from the general formula for rotational inertia, which is .5MR^2, where M is the mass of the object and R is the distance from the axis of rotation to the object's mass distribution. In the case of a cylinder, the mass distribution is uniform across the radius, so the distance R is simply the radius of the cylinder.

In conclusion, it is safe to assume that the solid cylinder and the cylindrical section have the same rotational inertia, as long as they have equal masses and radii. This is a fundamental principle in rotational dynamics and is applicable to various objects, not just cylinders.