Calculate the moment of inertia for the arrangement of seven disks

What is the moment of inertia of seven identical circular plane disks welded symmetrically?

Choose the correct answer:
1. 15mr²
2. 30mr²
3. 45mr²
4. 60mr²

Answer:

The correct answer is 30mr².

Moment of inertia is a key concept in rotational dynamics, especially when dealing with objects that have mass distributed along their axis. In this case, we have seven identical circular plane disks, each with mass (m) and radius (r), that are welded symmetrically to form the arrangement.

To calculate the moment of inertia for the entire arrangement about an axis normal to the plane and passing through point (P), we can use the parallel axis theorem. This theorem states that the total moment of inertia is the sum of the individual moment of inertia of each disk about its center of mass and the mass multiplied by the square of the distance from the center of mass to the axis of rotation.

For the central disk, the moment of inertia would be (1/2)mr² + mr² = 3/2*mr². As for the six surrounding disks, the moment of inertia for each would be (1/2)mr² + 4mr² = 9/2*mr² due to the distance between their center and point P being 2r.

By summing up the moment of inertia of the central disk and the six surrounding disks, we get a total moment of inertia for the entire arrangement of 30mr². This calculation demonstrates the application of the parallel axis theorem in determining the moment of inertia for complex configurations of rotating objects.