\(\mathrm{M}=\int_{0}^{\mathrm{L}}\left(\lambda_{0}+\mathrm{kx}\right) \mathrm{dx}\)

\(M=\lambda_{0} L+\frac{K \times L^{2}}{2}\)

\(\frac{2 \mathrm{M}-\lambda_{0} \mathrm{L}}{\mathrm{L}^{2}}=\mathrm{K}\)

\(\frac{2 \mathrm{M}}{\mathrm{L}^{2}}-\frac{\lambda_{0}}{\mathrm{L}}=\mathrm{K}\)

\(\frac{\int \mathrm{d} \mathrm{m}(\mathrm{r})}{\int \mathrm{d} \mathrm{m}}=\frac{\int(\lambda \mathrm{d} n) \mathrm{x}}{\mathrm{M}}=\frac{\int_{0}^{\mathrm{L}}\left(\lambda_{0} \mathrm{x}+\mathrm{k} \mathrm{x}^{2}\right) \mathrm{d} \mathrm{x}}{\mathrm{M}}\)

\(r_{c m}=\frac{\lambda_{0} L+\frac{k L^{2}}{2}}{M}\)

substitute \('k"\)

\(r_{c m}=\frac{2 L}{3}-\frac{\lambda_{0} \ell^{2}}{6 M}\)