If two identical disks are stacked on top of each other, what is the moment of inertia about the axis of the bottom disk?
Practice Questions
1 question
Q1
If two identical disks are stacked on top of each other, what is the moment of inertia about the axis of the bottom disk?
MR^2
2MR^2
1/2 MR^2
4MR^2
The moment of inertia of the bottom disk is MR^2, and the top disk contributes an additional MR^2 due to the parallel axis theorem, giving a total of 2MR^2.
Questions & Step-by-step Solutions
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Q
Q: If two identical disks are stacked on top of each other, what is the moment of inertia about the axis of the bottom disk?
Solution: The moment of inertia of the bottom disk is MR^2, and the top disk contributes an additional MR^2 due to the parallel axis theorem, giving a total of 2MR^2.
Steps: 10
Step 1: Understand that the moment of inertia (I) is a measure of how difficult it is to rotate an object around an axis.
Step 2: Identify that we have two identical disks, each with mass (M) and radius (R).
Step 3: Calculate the moment of inertia of the bottom disk about its own axis, which is given by the formula I = MR^2.
Step 4: Recognize that the top disk is also identical and has the same mass (M) and radius (R).
Step 5: Use the parallel axis theorem to find the moment of inertia of the top disk about the axis of the bottom disk. The parallel axis theorem states that I = I_cm + Md^2, where I_cm is the moment of inertia about its own center and d is the distance between the two axes.
Step 6: For the top disk, I_cm is also MR^2, and the distance (d) from the bottom disk's axis to the top disk's axis is equal to the diameter of the disk, which is 2R.
Step 7: Calculate the additional moment of inertia for the top disk: I_top = MR^2 + M(2R)^2 = MR^2 + 4MR^2 = 5MR^2.
Step 8: Add the moment of inertia of the bottom disk and the top disk: Total I = MR^2 (bottom disk) + 5MR^2 (top disk) = 6MR^2.
Step 9: However, we only need the contribution of the top disk to the bottom disk's moment of inertia, which is 4MR^2.
Step 10: Therefore, the total moment of inertia about the axis of the bottom disk is 2MR^2.
