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Question

In the figure shown below, a quarter ring of radius r is placed in the first quadrant of a cartesian co-ordinate system, with centre at origin. Find the co-ordinates of COM of the quarter ring.


A
(2Rπ,2Rπ)
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B
(2Rπ,0)
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C
(0,2Rπ)
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D
(0,0)
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Solution

The correct option is A (2Rπ,2Rπ)
Let us consider a small mass dm of elemental length ds which is at x m from y axis and y m from x axis.


Mass per unit length λ=dmds
dm=λds
dm=λrdθ (ds=rdθ)

To find COM:
xcm=xdmdm=π/20(rcosθ)(λrdθ)π/20λrdθ
=rπ/20cosθdθπ/20dθ=2Rπ
and ycm=ydmdm=π/20(rsinθ)(λrdθ)π/20λrdθ
=rπ/20sinθdθπ/20dθ=2Rπ
(xcm,ycm)=(2Rπ,2Rπ)

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