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 k X L n where n can be zero or any positive number. If the position x C M of the centre of mass of the rod is plotted against n , which of the following graphs best approximates the dependence of x C M on n ?

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 k X L n where n can be zero or any positive number. If the position x C M of the centre of mass of the rod is plotted against n , which of the following graphs best approximates the dependence of x C M on n ?, , , ,

MARKS App · Accepted Answer

x C · M = ∫ 0 L k L n · x n · dx x ∫ 0 L k L n · x n · dx ⇒ x C · M = ∫ 0 L x n + 1 dx ∫ 0 L x n dx = L n + 2 n + 2 · n + 1 L n + 1 ⇒ x C · M = L n + 1 n + 2 The variation of the mass with x is given by dx dn = L { n + 2 1 − n + 1 n + 2 2 } = L n + 2 2 If the rod has the same density as at x = 0 i.e., n = 0, therefore uniform, the cenre of mass would have been at L / 2 . As the density increases with length, the centre of mass shifts towards the right. Therefore it can only be (b).

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