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Question

A circular wire frame is made of four uniform quarter circular wires of same radius and of mass per unit length λ,2λ,3λ and 4λ as shown in figure. Assuming the origin to be at the centre of the circle, which of the following options is (are) correct?


  1. xcm=0
  2. ycm=0
  3. ycm<0
  4. xcm<0


Solution

The correct options are
A xcm=0
C ycm<0
Given, four uniform quarter circular rings of same radius and mass per unit length λ,2λ,3λ and 4λ.
If mass of ring with density λ=m, then
Mass of ring with density 2λ=2m
Mass of ring with density 3λ=3m
Mass of ring with density 4λ=4m
COM of quarter ring of mass m=(2Rπ,2Rπ)
COM of quarter ring of mass 2m=(2Rπ,2Rπ)
COM of quarter ring of mass 3m=(2Rπ,2Rπ)
COM of quarter ring of mass 4m=(2Rπ,2Rπ)

To find COM of system
xcm=m1x1+m2x2+m3x3+m4x4m1+m2+m3+m4
=m(2Rπ)+2m(2Rπ)+3m(2Rπ)+4m(2Rπ)10m
=2Rπ(m2m3m+4m)10m
=0
ycm=m1y1+m2y2+m3y3+m4y4m1+m2+m3+m4
=m(2Rπ)+2m(2Rπ)+3m(2Rπ)+4m(2Rπ)10m
=2Rπ(m+2m3m4m)10m
=4R5π<0
Option (a) and (c) are correct.

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