Let rod is placed along $ x- $ axis. Mass of element PQ of length $ dx $ situated at $ x=x $ is
$ dm=\lambda dx=(2+x)dx $ The COM of the element has coordinates $ (x,0,0). $ Therefore, $ x- $ coordinate of COM of the rod will be $ {{x}_{COM}}=\frac{\int_\limits{0}^{3}{xdm}}{\int_\limits{0}^{3}{dm}} $ $ =\frac{\int_\limits{0}^{3}{x(2+x)dx}}{\int_\limits{0}^{3}{(2+x)dx}} $ $ =\frac{\int_\limits{0}^{3}{(2x+{{x}^{2}})dx}}{\int_\limits{0}^{3}{(2+x)dx}} $ $ =\frac{\left[ \frac{2{{x}^{2}}}{2}+\frac{{{x}^{3}}}{3} \right]_{0}^{3}}{\left[ 2x+\frac{{{x}^{2}}}{2} \right]_{0}^{3}} $ $ =\frac{\left[ {{(3)}^{2}}+\frac{{{(3)}^{3}}}{3} \right]}{\left[ 2\times 3+\frac{{{(3)}^{2}}}{2} \right]} $ $ =\frac{9+9}{6+9/2} $ $ =\frac{18\times 2}{21} $ $ =\frac{12}{7}m $
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Top Questions on System of Particles & Rotational Motion
The system of particles refers to the extended body which is considered a rigid body most of the time for simple or easy understanding. A rigid body is a body with a perfectly definite and unchangeable shape.
The distance between the pair of particles in such a body does not replace or alter. Rotational motion can be described as the motion of a rigid body originates in such a manner that all of its particles move in a circle about an axis with a common angular velocity.
The few common examples of rotational motion are the motion of the blade of a windmill and periodic motion.
