Let S be the system of point masses p1, ...., pk, and let T be the system of poi

Question

Let S be the system of point masses p1, ...., pk, and let T be the system of point masses pk+1, ...., pn.Suppose that each point mass, pi, has mass mi at that each that each point pilies on the real line and has x-coordinate equal to xi. Prove that the center of mass of thesystem S U T of consisting of all of the point masses,p1, ...., pk,, is equal to the center of mass of the system consisting of two specialpoint masses: one point mass pS which has a mass equal to the totalmass of the system S and which has x-coordinate equal to the centerof mass of the system S and a second point mass pT denedanalogously with respect to the system T.

Explanation / Answer

let suppose... Total inertia I = sum of the 3 contributing inertias. Define I1 as inertia of each of two nearest masses, I2 as inertia of diagonally opposite mass, and r as the side of the square. The length of the diagonal = sqrt(2)r. I = 2I1+I2 = 2mr^2+m(sqrt(2)r)^2 = 4mr^2 KE = Iw^2/2 ==> w = sqrt(2KE/I) RPM = w/(2pi*60)

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