b
સળીયાની રેખીય ઘનતા અંતર સાથે \(m/dx = \lambda\) \(dm = \lambda dx\) મુજબ ઘટે છે

દ્રવ્યમાન કેન્દ્રનું સ્થાન \({x_{cm}}\,\, = \,\,\frac{{\int {dm\, \times \,x} }}{{\int {dm} }}\)

\( = \,\,\frac{{\int\limits_0^3 {(\lambda \,dx)\, \times \,x} }}{{\int\limits_0^3 {\lambda \,dx} }}\,\, = \,\,\frac{{\int\limits_0^3 {(2 + x)\, \times \,xdx} }}{{\int\limits_0^3 {(2 + x)\,\,dx} }}\)

\(\, = \,\,\frac{{\left[ {{x^2}\, + \,\frac{{{x^3}}}{3}} \right]_0^3}}{{\left[ {2x\, + \,\frac{{{x^2}}}{2}} \right]_0^3}}\,\,\, = \,\,\frac{{9\, + \,9}}{{6\, + \,\frac{9}{2}}}\,\, = \,\,\frac{{36}}{{21}}\,\, = \,\,\frac{{12}}{7}\,m\)