4.3 Center of Mass and Motion of a System

• The center of mass of a system is the unique point where the weighted average position of all mass
• The formula for calculating the center of mass is also expressed as x_CM
• The center of mass formula simplifies the analysis of a system's motion by treating all mass as concentrated at a single point.


  True
• The formula for calculating the center of mass is x_CM
• What does $x_{i}$ represent in the center of mass formula?


  Position of each particle
• What are the two parameters that need to be identified to calculate the center of mass in a system?

  Mass and position
• The unit of center of mass position is meters (m).


  True
• The center of mass of the example system is located at 2.8 meters.
• The formula for calculating the center of mass is the same for both one-dimensional and two-dimensional systems.

  False
• Two particles with equal mass and positions have their center of mass at the midpoint.

  True
• The center of mass of the second example system is located at 2.8 meters.
• The final step in calculating the center of mass is to analyze the data.

  False
• What does the formula $\frac{\sum m_{i} y_{i}}{\sum m_{i}}$ represent?


  y-coordinate of CM
• To apply the center of mass formula for two-dimensional systems, we need to calculate the coordinates independently.
• The <span class="tex-inline">x_{CM}</span> and <span class="tex-inline">y_{CM}</span> represent the coordinates of the center of mass.


  True
• Match the dimensionality with the corresponding center of mass formula:

  One-Dimensional ↔️ $x_{CM} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$

  Two-Dimensional ↔️ $x_{CM} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$ and $y_{CM} =$$\frac{\sum m_{i} y_{i}}{\sum m_{i}}$
• What are the mass and coordinates of Particle 1 in the example calculation?

  $m_{1} =$$2 \, \text{kg}$, x_1 = 1 \, \text{m}</latex>, $y_{1} =$$2 \, \text{m}$
• The center of mass acts as if all the mass of the system is concentrated at that single point.


  True
• Why is focusing on the motion of the center of mass useful in understanding a complex system?

  Simplifies overall motion analysis
• The center of mass acts as if all the mass of the system is located there.


  True
• The formula for calculating the center of mass is x_CM
• What does $m_{i}$ represent in the center of mass formula?


  Mass of each particle
• $x_{i}$ represents the position of each particle in the center of mass formula.


  True
• Two particles of masses 2 kg and 3 kg are located at 1 m and 4 m, respectively. The center of mass is 2.8
• What are the units of the center of mass position ($x_{CM}$)?


  meters
• What does $m_{i}$ represent in the center of mass formula?


  Mass of each particle
• The center of mass is the weighted average position of all mass in a system.

  True
• Steps to apply the center of mass formula for a one-dimensional system:

  1️⃣ Identify the masses and positions of all particles.

  2️⃣ Plug the values into the center of mass formula.
• What is the formula for the center of mass in a one-dimensional system?

  x_{CM} = \frac{\sum m_i x_i}{\sum m_i}</latex>
• The units of the center of mass position ($x_{CM}$) are meters.


  True
• Consider a system with two particles: 2 kg at 1 m and 3 kg at 4 m. The center of mass is 2.8
• The center of mass of a one-dimensional system simplifies the analysis of its motion.


  True
• The formula for calculating the center of mass position in a one-dimensional system is x_{CM} = \frac{\sum m_i x_i}{\sum m_i}
• In the example calculation, what are the mass and position of Particle 1?

  m_1 = 2 \, \text{kg}</latex>, $x_{1} =$$1 \, \text{m}$
• The center of mass formula for one-dimensional systems includes the sum of the product of mass and position divided by the total mass.


  True
• What is the definition of the center of mass in a system?

  Weighted average position of mass
• The center of mass acts as if all the mass of the system is located at a single point.
• Match the variable with its definition:

  $x_{CM}$ ↔️ Center of mass position

  $m_{i}$ ↔️ Mass of each particle

  $x_{i}$ ↔️ Position of each particle
• What is the unit of mass in the center of mass formula?

  Kilograms (kg)
• The center of mass for the two-particle system in the example is calculated to be 2.8 meters.