Question

4 particles of mass $2\text{kg}$ each are placed on the $X$-axis at positions $X_1=2,X_2=4,X_3=6$ and $X_4=8$. If another mass of $m\text{kg}$ is placed on $X$-axis at $x=a$, choose the correct statements.

• A. If $m=4\text{kg},a=8$, $COM$ will move by $2\text{units}$ distance.
• B. If $m=4\text{kg},a=8$, $COM$ will move by $1\text{unit}$ distance.
• C. If $a=5$, $COM$ will not move by any distance.
• D. If $a=5$, $COM$ will move by a distance which depends on the value of $m$.

Answer 1

Answer: (B), (C)

The correct options are
B If $m=4\text{kg},a=8$, $COM$ will move by $1\text{unit}$ distance.
C If $a=5$, $COM$ will not move by any distance.

Initial position of $COM$,
$X_{COM}=\frac{M_1{X}_1+M_2{X}_2+M_3{X}_3+M_4{X}_4}{M_1+M_2+M_3+M_4}$
$=\frac{2\times 2+2\times 4+2\times 6+2\times 8}{2+2+2+2}$
$X_{COM}=5$

When $m=4\text{kg}$ is placed at $X=8$
$X_{COM}^{\prime }=\frac{2\times 2+2\times 4+2\times 6+2\times 8+4\times 8}{2+2+2+2+4}=\frac{72}{12}$
$X_{COM}^{\prime }=6$

Therefore, $COM$ will be displaced by distance
$d=6-5=1$ unit

We have initial coordinate of $COM$ is $X=5$, therefore any mass placed at $X=5$ will not change the position of $COM$ because $COM$ of both (system + new mass) will be at the same point. And it will not depend on the mass of the new object.