2. The following principle is used frequently in physics: if system S of point m

Question

2. The following principle is used frequently in physics: if system S of point masses is given, then we can regard the system as a single point mass located at the center of mass of S and having mass equal to the total mass of S. For example, a large asymmetric object might, for simplicity, be regarded as only occupying a single point in space and that the mass of this large object might be regarded as concentrated at this single point. The natural candidate for such a point is the center of mass of the object. In this exercise, you will provide a mathematical justication for why this is "natural"

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 andhas 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,, isequal 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

Considering system S U T centre of mass = ? xi * mi/? mi for i = 1 to n i.e, centre of mass = (x1 * m1 + x2* m2 + ... xn * mn)/(m1 + m2 + ... + mn) Centre of mass of system S centre of mass = ? xi * mi/? mi for i = 1 to k i.e, centre of mass of system S = (x1 * m1 + x2* m2 + ... xk * mk)/(m1 + m2 + ... + mk) -- EQ (1) Centre of mass of system T centre of mass = ? xi * mi/? mi for i = k + 1 to n i.e, centre of mass of system S = (x(k+1) * m(k+1) + x(k+2)* m(k+2) + ... xn * mn)/(m(k+1) + m(k+2) + ... + mn) Now we have two point masses where Ps = m1 + m2 + ... + mk Pt = mk+1 + mk+2 + .... +mn and their poistions are also known Centre of mass of two point masses = Centreof mass of system S * Ps + Centreof mass of system T * Pt /(Ps + Pt) = ((x1 * m1 + x2* m2 + ... xk * mk)/(m1 + m2 + ... + mk) * (m1 + m2 + ... + mk) + (x(k+1) * m(k+1) + x(k+2)* m(k+2) + ... xn * mn)/(m(k+1) + m(k+2) + ... + mn) * (mk+1 + mk+2 + .... +mn))/ ( m1 + m2 +... + mk + mk+1 + ... + mn) = (x1 * m1 + x2* m2 ... xk * mk + x(k+1) * m(k+1) + x(k+2)* m(k+2) + ... + xn * mn)/(m1 + m2 + ... mn) which is same as above EQ(1) Hence proved

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