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Question

# A thin rod of length L is lying along the x− axis with its ends at x=0 and x=L. Its linear density (mass/length) varies with x as λ=k(xL)n, when n can be zero or any positive number. If the position xcm of the centre of mass of the rod is plotted against ′n′, which of the following graphs best approximates the dependence of xcm on n?

A
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B
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C
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D
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Solution

## The correct option is D When n=0, λ=k where k is a constant. It means that the linear mass density is constant. In this case, the centre of mass will be at the middle of the rod. i.e. at L2 Now, applying the fundamental equation for COM of system, xcm=∫L0x dm∫L0dm=∫L0x λ dx∫L0λ dx =∫L0kx(xL)ndx∫L0k(xL)ndx=k.[xn+2(n+2)Ln]L0[kxn+1(n+1)Ln]L0 ⇒xcm=L(n+2)L(n+1)×(n+1)(n+2) ∴xcm=L(n+1)(n+2) Putting n=0,xcm=L2 Hence a & b options are incorrect) For n=1;xcm=2L3 For n=2,xcm=3L4 Finding the value of xcm when n approaches ∞ limn→∞xcm=limn→∞L(1+1n)(1+2n) ⇒limn→∞xcm=L(1+0)(1+0)=L Slope of xcm vs n is given by: dxcmdn=L[(n+2).1−(n+1).1(n+2)2] dxcmdn=L(n+2)2 ⇒The slope is always +ve, which indicates that the curve of xcm vs n follows a continously increasing trend till n→∞ ∴ option (d) is correct.

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