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Q.

The center of mass of a non uniform rod of Length L whose mass per unit length $λ = \frac{K x^{2}}{L}$ , where K is constant and x is the distance from one end is

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a

$\frac{K}{L}$

b

$\frac{3 L}{4}$

c

$\frac{3 K}{L}$

d

$\frac{L}{8}$

answer is A.

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Detailed Solution

Question Image

$λ = \frac{k x^{2}}{L}$

$dm = λ d x$

$x_{cm} = \frac{\int_{0}^{L} d m x}{\int_{0}^{L} d m}$

$= \frac{\int_{0}^{L} \frac{k x^{2}}{L} . d x . x}{\int_{0}^{L} \frac{k x^{2}}{L} . d x} = \frac{\int_{0}^{L} x^{3} d x}{\int_{0}^{L} x^{2} d x}$

$= \frac{{[\frac{x^{4}}{4}]}_{0}^{L}}{{[\frac{x^{3}}{3}]}_{0}^{L}} = \frac{3}{4} \frac{L^{4}}{L^{3}} = \frac{3}{4} L$

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