A machine part is made from a uniform solid disk of radius R and mass M. A hole of radius R/2 is drilled into the disk, with the center of the hole at a distance R/2 from the center of the disk (the diameter of the hole spans from the center of the disk to its outer edge). What is the moment of inertia of this machine part about the center of the disk in terms of R and M? (Use any variable or symbol stated above as necessary.)

Question

A machine part is made from a uniform solid disk of radius R and mass M. A hole of radius R/2 is drilled into the disk, with the center of the hole at a distance R/2 from the center of the disk (the diameter of the hole spans from the center of the disk to its outer edge). What is the moment of inertia of this machine part about the center of the disk in terms of R and M? (Use any variable or symbol stated above as necessary.)

I =
I need to know the formula or equation for this problem. No calculations needed.

Answer 1

To calculate the moment of inertia of the machine part about the center of the disk, we can use the parallel axis theorem. This theorem states that the moment of inertia of an object about any axis parallel to an axis passing through its center of mass is equal to the sum of its moment of inertia about the center of mass and the product of its mass and the square of the distance between the two axes.

In this case, the moment of inertia of the machine part about the axis passing through its center of mass can be calculated using the moment of inertia formula for a solid disk:

I_disk = (1/2) * M * R^2

where I_disk is the moment of inertia of the solid disk, M is the mass of the disk, and R is the radius of the disk.

Since there is a hole in the disk, we need to subtract the moment of inertia of the material removed. The moment of inertia of a solid disk with a hole can be calculated as the difference between the moment of inertia of the full disk and the moment of inertia of the removed material. The removed material can be considered as a small disk with radius R/2 and mass M' (the mass of the removed material).

The moment of inertia of the removed material is given by:

I_hole = (1/2) * M' * (R/2)^2

Now, we can use the parallel axis theorem to calculate the moment of inertia of the machine part about the center of the disk. The distance between the two axes is R/2, as given in the problem.

Therefore, the moment of inertia of the machine part about the center of the disk, I, can be calculated as:

I = I_disk - I_hole + M' * (R/2)^2

Simplifying this equation will give you the moment of inertia of the machine part in terms of R and M.