Question

Find the coordinates of Center of a mass of a non-uniform rod which has liner mass density $\lambda =4+BX$ whose total length is L.

Answer 1

Consider a differential part of the rod, then

At difference $x$ from $o$

$\alpha =\frac{dm}{dx}=4+Bx$

$dm=(4+Bx).dx$

Lett the distance of $COH$ from $O$ be $x_{COM}$

Then,

$x_{COM}=\frac{{\int }_O^{M}dmx}{{\int }_o^{M}dm}$

$x_{COM}=\frac{{\int }_O^L(4+Bx)\infty dr}{{\int }_O^L(1+Bx)dr}$

$x_{COM}=\frac{{\int }_O^L(4x+\frac{B{x}^2}{2})dr}{{\int }_O^L(4+bx)dr}$

$x_{COM}=\frac{{[2x^2+\frac{B{x}^3}{6}]}_0^2}{{[4x+\frac{B{x}^2}{2}]}_0^2}$

$x_{COM}=\frac{2L^2+\frac{B{L}^3}{6}}{4L+\frac{B{L}^2}{2}}$

$x_{COM}=\left(\frac{12+BL}{24+3BL}\right)$