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Question

The center of mass of a non-uniform rod of length L=2 m, whose mass per unit length varies as λ=Kx2L where K is a positive constant and x is the distance of any point from the left end of the rod, lies at a distance in metre of (from the left end) is

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Solution


Mass per unit length of the rod, (λ)=Kx2L

Let us consider an element of length dx located at a distance x away from the left end of the rod and its mass is,

dm=λdx

dm=Kx2L×dx

The x-coordinate of center of mass of the rod is,

xcom=xdmdm

Integrating with proper limits we get,

xcom=L0x×Kx2LdxL0Kx2Ldx

xcom=L34L23=3L4

xcom=3×24=1.5 m

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