Question

conider the following familiar system in polar coordinates. let d be the disk. is d an invariant set? ndoes d attract an open set of initial conditions

Answer 1

To ensure that System B has a moment of inertia equal to System A, System B must contain four smaller disks when the larger disks' radius is R and the smaller disks' radius is 2R.

Step-by-step explanation:

To compare the moment of inertia between System A and System B, we first need to understand that the moment of inertia (I) for a single disk with mass M and radius R is given by the formula I = ½MR². For a disk with radius 2R, the moment of inertia would be I = ½M(2R)² = 2MR². System A has two larger disks, so its total moment of inertia is 2 * 2MR² = 4MR². System B has one larger disk (2MR²) and n smaller disks (each with moment of inertia ½MR²).

For the moments of inertia to be equal, the total moment of inertia for System B must equal that of System A, which means:

2MR² + n(½MR²) = 4MR²

Solving for n gives:

2MR² + ½nMR² = 4MR²

½nMR² = 2MR²

n = 4

Therefore, System B must contain four smaller disks to have a moment of inertia equal to that of System A.