Moment of Inertia of Disk with Off-Center Hole

[Quartz]

A curcial part of a piece of machinery starts as a flat uniform cylindrical disk of radius R0 and mass M. It then has a circular hole of radius R1 drilled into it. The hole's center is a distance h from the center of the disk. Find the moment of inertia of this disk (with off-center hole) when rotated about its center, C.

Hint: Consider a solid disk and subtract the hole; use parallel-axis theorem.

 

[rohanprabhu]

firstly.. for all that i know, this is a post that belongs in the 'Homework and Help forum'. Secondly, you need to show some efforts from your side in solving this problem before we can provide you any help with this question.

 

[Moetasim]

The moment of inertia of a disk with an off-center hole can be calculated by considering it as a solid disk and subtracting the moment of inertia of the hole using the parallel-axis theorem. This theorem states that the moment of inertia of a body about a certain axis is equal to the moment of inertia about a parallel axis passing through the center of mass, plus the product of the mass of the body and the square of the distance between the two axes.

In this case, we can consider the disk as a combination of two parts: the solid disk with radius R0 and the hole with radius R1. The moment of inertia of the solid disk can be calculated using the formula for a solid cylinder, which is (1/2)MR0^2. The moment of inertia of the hole can be calculated using the formula for a hollow cylinder, which is (1/2)M(R1^2 + R0^2).

Since the hole is off-center, we need to use the parallel-axis theorem to find the moment of inertia of the hole about the center of the disk. This can be done by adding the product of the mass of the hole and the square of the distance between the two axes, which is equal to (1/2)Mh^2.

Therefore, the moment of inertia of the disk with off-center hole can be calculated as (1/2)MR0^2 - (1/2)M(R1^2 + R0^2) + (1/2)Mh^2. This can also be written as (1/2)M(R0^2 - R1^2 + h^2), which is the final expression for the moment of inertia.

In conclusion, the moment of inertia of a disk with an off-center hole can be calculated by considering it as a combination of two parts and using the parallel-axis theorem. This calculation is crucial for understanding the rotational dynamics of the disk and designing machinery that utilizes it.