Question

The mass of particles is 100 g and 300 g. At a given time have velocities are $10\hat{i}-7\hat{j}-3\hat{k}$ and $7\hat{i}-9\hat{j}+6\hat{k}$ respectively. Determine velocity of center of mass.

Answer 1

The velocity of the center of mass

1. A change in an object's position over time is called velocity.
2. The velocity of the center of mass describes the overall movement since the center of mass moves along with an object or system of particles in motion.

Calculation of velocity of the center of mass

Step 1: Given parameter

$$\begin{gathered} \mathrm{Mass}\mathrm{of}\mathrm{particle}1\left({\mathrm{m}}_1\right)=100\mathrm{g} \\ 1g=0.001\mathrm{kg} \\ 100g=100\times 0.001 \\ =0.1\mathrm{kg} \\ \mathrm{Mass}\mathrm{of}\mathrm{particle}2\left({\mathrm{m}}_2\right)=300\mathrm{g} \\ 300g=300\times 0.001 \\ =0.3\mathrm{kg} \\ \mathrm{Velocity}\mathrm{of}\mathrm{particle}1\left({\mathrm{v}}_1\right)=10\hat{\mathrm{i}}-7\hat{\mathrm{j}}-3\hat{\mathrm{k}} \\ \mathrm{Velocity}\mathrm{of}\mathrm{particle}2\left({\mathrm{v}}_2\right)=7\hat{\mathrm{i}}-9\hat{\mathrm{j}}+6\hat{\mathrm{k}} \end{gathered}$$

Step 2: The velocity of the center of mass

$\overrightarrow{\mathrm{V}}={\overrightarrow{\mathrm{V}}}_x+{\overrightarrow{\mathrm{V}}}_y+{\overrightarrow{\mathrm{V}}}_z$

$$\begin{gathered} {\overrightarrow{\mathrm{V}}}_x-\mathrm{Velocity}\mathrm{of}\mathrm{center}\mathrm{of}\mathrm{mass}\mathrm{in}\mathrm{x}-\mathrm{direction} \\ {\overrightarrow{\mathrm{V}}}_y-\mathrm{Velocity}\mathrm{of}\mathrm{center}\mathrm{of}\mathrm{mass}\mathrm{in}y-\mathrm{direction} \\ {\overrightarrow{\mathrm{V}}}_z-\mathrm{Velocity}\mathrm{of}\mathrm{center}\mathrm{of}\mathrm{mass}\mathrm{in}z-\mathrm{direction} \end{gathered}$$

Step 3: Velocity of the center of mass in any direction

$\overrightarrow{\mathrm{V}}=\sum _{\mathrm{i}=1}^2\frac{{\mathrm{m}}_{\mathrm{i}}{\mathrm{v}}_{\mathrm{i}}}{\mathrm{M}}$

${\mathrm{m}}_{\mathrm{i}}{\mathrm{v}}_{\mathrm{i}}$is the sum of the momentum.

$\mathrm{M}$ is the total mass of particles.

Step 4: To calculate ${\mathrm{V}}_{\mathrm{x}},{\mathrm{V}}_{\mathrm{y}}\mathrm{and}{\mathrm{V}}_{\mathrm{z}}$

$$\begin{gathered} \overrightarrow{{\mathrm{V}}_{\mathrm{x}}}=\frac{0.1\times 10+0.3\times 7}{0.4} \\ =\frac{31}{4}\mathrm{m}/\mathrm{s} \\ \overrightarrow{{\mathrm{V}}_{\mathrm{y}}}=\frac{0.1\times (-7)+0.3\times (-9)}{0.4} \\ =-\frac{34}{4}\mathrm{m}/\mathrm{s} \\ \overrightarrow{{\mathrm{V}}_{\mathrm{z}}}=\frac{0.1\times (-3)+0.36}{0.4} \\ =\frac{15}{4}\mathrm{m}/\mathrm{s} \end{gathered}$$

Step 5: Total velocity of center of mass

$\overrightarrow{\mathrm{V}}=\frac{31}{4}-\frac{34}{4}+\frac{15}{4}=3\mathrm{m}/\mathrm{s}.$

Therefore, the velocity of the center of mass is $3\mathrm{m}/\mathrm{s}$.