Question

A non-uniform thin rod of length $L$ is placed along $x-$axis as such its one of ends at the origin. The linear mass density of rod is $\lambda ={\lambda }_{0}x$. The distance of centre of mass of rod from the origin is:

• A. $L/2$
• B. $2L/3$
• C. $L/4$
• D. $L/5$

Answer 1

The correct option is B $2L/3$

Given that,

Density of rod $\lambda ={\lambda }_{0}x$

Length of rod = $L$

Now, center of mass

$C.M=\frac{1}{M}\underset{0}{\overset{L}{\int }}xdm$

Now, total mass

$M=\underset{0}{\overset{L}{\int }}dm$

$M=\underset{0}{\overset{L}{\int }}{\lambda }_{0}dx$

$M={\lambda }_0\underset{0}{\overset{L}{\int }}xdx$

$M={\lambda }_0{\left[\frac{x^2}{2}\right]}_0^L$

$M=\frac{{\lambda }_0{L}^2}{2}....\left(I\right)$

Now, center of mass

$C.M=\frac{1}{M}\underset{0}{\overset{L}{\int }}xdm$

$C.M=\frac{1}{M}\underset{0}{\overset{L}{\int }}x\left(\lambda dx\right)$

$C.M=\frac{1}{M}{\lambda }_0\underset{0}{\overset{L}{\int }}x\left(xdx\right)$

$C.M=\frac{1}{M}{\lambda }_0{\left[\frac{x^3}{3}\right]}_0^L$

$C.M=\frac{1}{M}{\lambda }_0\frac{L^3}{3}$

Now, put the value of $\frac{{\lambda }_0}{M}$from equation (I)

$C.M=\frac{1}{M}{\lambda }_0\frac{L^3}{3}$

$C.M=\frac{L^3}{3}\times \frac{2}{L^2}$

$C.M=\frac{2L}{3}$

Now, the distance of center of mass of rod from the origin

Hence, The distance of center of mass of rod from the origin is $\frac{2L}{3}$