Question

As shown in diagram there are five identical rods. Length of each rod is $l$ and mass $m$. Find out distance of $\text{C.O.M.}$ of system From $\left(C\right)$

Answer 1

Given:
Length of each rod $=l$
Mass of each rod $=m$

$\text{COM}$ of system of particles is given by ${\overrightarrow{r}}_{\text{COM}}=\frac{m_1{\overrightarrow{r}}_1+m_2{\overrightarrow{r}}_2+m_3{\overrightarrow{r}}_3+m_4{\overrightarrow{r}}_4\cdots \cdots }{m_1+m_2+m_3+m_4+m_5+\dots }$

${\overrightarrow{r}}_{\text{COM}}=\frac{m(\frac{l}{2}\hat{i}+0j)+m(0\hat{i}+\frac{1}{2}j)+m(\frac{l}{2}\hat{i}+j)+m(0\hat{i}+\frac{3}{2}lj)+m(\frac{1}{2}\hat{i}+2lj)}{m+m+m+m+m}$

${\overrightarrow{r}}_{COM}=(\frac{3l}{10}\hat{i}+l\hat{j})$

Now the magnitude of $\text{COM}$
$\left|{\overrightarrow{r}}_{\text{COM}}\right|=\sqrt{{\left(\frac{3l}{10}\right)}^2+l^2}$

$\left|{\overrightarrow{r}}_{\text{COM}}\right|=\sqrt{\frac{109}{100}}l$

Hence , option $\left(D\right)$ is correct.