Moment of Inertia of a Disk

In summary, the conversation discusses the calculation of the new rotational inertia of a disk with a drilled hole at the center. The solution involves using the parallel-axis theorem and subtracting the moment of inertia of the missing piece from that of the whole disk. It is found that the correct answer should be (1/2)MR2 - (3/512)MR2, taking into account the moment of inertia of the drilled out piece.
  • #1
Jefffff
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Homework Statement


A disk of radius R has an initial mass M. Then a hole of radius (1/4) is drilled, with its edge at the disk center.

Find the new rotational inertia about the central axis. Hint: Find the rotational inertia of the missing piece, and subtract it from that of the whole disk. You’ll find the parallel-axis theorem helpful.
Express your answer in terms of the variables M and R.

Homework Equations


I = Icm + Md2

Moment of Inertia of a Disk: 1/2 MR2

The Attempt at a Solution



Based on the formula for mass of the disk, let M' be mass of the drilled out smaller disk.

M = 2πr2
Since all factors are constant here except r, where the new radius is 1/4 of the original, thus (1/4)2 = (1/16) M

so M' = (1/16)M

Plugging in values to find the moment of inertia of the drilled out disk piece:

I = Icm + Md2

I = (1/2)(1/16M)(R/4)2 + (M/16)(R/4)2

This yields I = (3/512) MR2

The new moment of inertia is (1/2)MR2 - (3/512)MR2

= (253/512) MR2

The problem is the correct answer for this question according to the quiz was:

(1/2)MR - (1/256)MR2

which is different from what I got. Not sure where I went wrong. Anything helps!
 
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  • #2
It seems to me that the answer from the quiz fails to account for the hole’s centre of mass moment of inertia.
 
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  • #3
Orodruin said:
It seems to me that the answer from the quiz fails to account for the hole’s centre of mass moment of inertia.
Yes, this is what seems to be the case. Thanks for checking the work! I'll submit a request to the instructor.
 

What is moment of inertia?

The moment of inertia of an object is a measure of its resistance to rotational motion. It is a property of an object that depends on its mass distribution and the axis of rotation.

How is moment of inertia calculated for a disk?

The moment of inertia for a disk can be calculated using the formula I = 1/2 * m * r^2, where m is the mass of the disk and r is the radius of the disk.

Why is moment of inertia important for a disk?

Moment of inertia is important for a disk because it affects how the disk will rotate when a force is applied to it. A larger moment of inertia means the disk will require more torque to rotate, while a smaller moment of inertia means it will rotate more easily.

How does the moment of inertia change for a disk with a different mass or radius?

The moment of inertia of a disk is directly proportional to its mass and the square of its radius. This means that as the mass or radius of the disk increases, the moment of inertia will also increase.

Can the moment of inertia of a disk be affected by its shape or thickness?

Yes, the shape and thickness of a disk can affect its moment of inertia. For example, a disk with a greater thickness will have a larger moment of inertia than a disk with the same mass and radius but a smaller thickness. Similarly, a disk with a non-uniform mass distribution will have a different moment of inertia than a disk with a uniform mass distribution.

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