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Question

A square frame is made using four thin uniform rods of length l each having linear mass densities λ, 2λ, 3λ and 4λ as shown in the figure. Which of the following is correct?

A
xcm=0,ycm=0
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B
xcm>0,ycm>0
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C
xcm<0,ycm<0
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D
xcm=0,ycm<0
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Solution

The correct option is D xcm=0,ycm<0Mass of rods will be m1=λl, m2=2λl, m3=3λl, m4=4λl whose centre of mass are shown in the figure given below :- Position of x coordinate of centre of mass : xcm=m1x1+m2x2+m3x3+m4x4m1+m2+m3+m4=λl(l2√2)+2λl(−l2√2)+3λl(−l2√2)+4λl(l2√2)λl+2λl+3λl+4λl =12√2(l−2l−3l+4l)10=0ycm=m1y1+m2y2+m3y3+m4y4m1+m2+m3+m4=λl(l2√2)+2λl(l2√2)+3λl(−l2√2)+4λl(−l2√2)10λl =12√2(l+2l−3l−4l)10=−l5√2 So, xcm=0 and ycm<0

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