Moment of Inertia of a disk (question on formula involving radius)

In summary, the moment of inertia for a disk with two radii can be calculated using the parallel axis theorem by breaking the disk into separate parts and adding the moments of inertia of each part. The final equation is (M * (A^2 + B^2))/2.
  • #1
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Homework Statement


Determine the moment of inertia of a think disk of mass M, with inner radius A and outer radius B, when rotating about an axis perpendicular to the plane of the disk and through its center.

Homework Equations


The equation of moment of inertia for a disk is I = (m*r^2)/2
my question is that because there are two radii, how does that affect r in the equation?

The Attempt at a Solution


I = (m*r^2)/2 then substitute I = (M*r^2)/2 but is r=(B-A)^2 or is r=(B^2-A^2)?
 
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  • #2


Hello,

Thank you for your question. The moment of inertia for a disk with two radii can be calculated by breaking the disk into separate parts and using the parallel axis theorem.

First, we can calculate the moment of inertia for the inner disk with radius A:

I1 = (m1 * r1^2)/2 = (M * A^2)/2

Next, we can calculate the moment of inertia for the outer ring with radius B:

I2 = (m2 * r2^2)/2 = (M * (B-A)^2)/2

Then, we can use the parallel axis theorem to calculate the moment of inertia for the entire disk:

I = I1 + I2 + M * d^2

Where d is the distance between the axis of rotation and the center of mass of the disk, which in this case is equal to (A+B)/2. Therefore, the final equation for the moment of inertia of a disk with two radii is:

I = (M * A^2)/2 + (M * (B-A)^2)/2 + M * ((A+B)/2)^2

I = (M * (A^2 + (B-A)^2 + (A+B)^2))/2

I = (M * (A^2 + B^2))/2

I = (M * (A^2 + B^2))/2

I = (M * (A^2 + B^2))/2

I = (M * (A^2 + B^2))/2

Therefore, the moment of inertia of a disk with two radii is (M * (A^2 + B^2))/2. I hope this helps clarify your question. Let me know if you have any further questions. Good luck with your calculations!
 

What is the formula for calculating the moment of inertia of a disk?

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

Why is the moment of inertia of a disk important?

The moment of inertia of a disk is important because it helps us understand how an object will respond to external forces, such as rotation or angular acceleration. It is also used in various engineering applications, such as designing machines and calculating the stability of structures.

How does the radius of a disk affect its moment of inertia?

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

Is the moment of inertia of a disk affected by its mass?

Yes, the moment of inertia of a disk is directly proportional to its mass. This means that as the mass of a disk increases, its moment of inertia also increases.

Can the moment of inertia of a disk change?

Yes, the moment of inertia of a disk can change if its mass or radius changes. It can also change if the disk experiences external forces that cause it to rotate or accelerate angularly.

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