2.1 Systems and Center of Mass

• What is a system in physics?

  Object or set of objects
• The center of mass for a single object is its geometric center.
• Match the variables in the center of mass formula with their definitions:

  $\vec{r}_{CM}$ ↔️ Position vector of the center of mass

  $m_{i}$ ↔️ Mass of the i-th particle

  $\vec{r}_{i}$ ↔️ Position vector of the i-th particle
• Defining a system helps identify its boundaries and interactions.
• The center of mass allows a system to be treated as a single point when analyzing its motion.


  True
• Compare the properties of the center of mass for a single object versus a system of particles:

  1️⃣ For a single object, the CM is the geometric center

  2️⃣ For a system of particles, the CM is calculated using the formula

  3️⃣ The entire object is treated as a point for analysis
• The center of mass simplifies the analysis of system motion by treating it as a single point object.

  True
• What does $\vec{r}_{CM}$ represent in the formula for the center of mass?


  Position vector of CM
• The center of mass allows us to treat a system as a single point object when analyzing its motion.

  True
• What is a discrete mass distribution?

  Distinct particles or objects
• Understanding the center of mass simplifies the application of Newton's laws of motion.


  True
• In the general formula for continuous mass distributions, the numerator involves the integral of $\vec{r}$ and dm
• Steps to calculate the center of mass for a continuous mass distribution in one dimension:

  1️⃣ Define the linear mass density $\lambda(x)$

  2️⃣ Calculate the total mass $M =$$\int \lambda(x) \, dx$

  3️⃣ Compute the CM position $x_{CM} =$$\frac{\int x \lambda(x) \, dx}{M}$
• For a single object, the center of mass is located at its geometric center.


  True
• The center of mass (CM) is the point where the total mass of an object or system is concentrated
• What is the formula for calculating the center of mass for a system of particles?

  $\vec{r}_{CM} =$$\frac{\sum m_{i} \vec{r}_{i}}{\sum m_{i}}$
• The center of mass for a continuous mass distribution is calculated using an integral formula.


  True
• What is Newton's Second Law of Motion?

  $\vec{F}_{net} =$$m\vec{a}$
• What is the formula for the net force on a system of particles in terms of the center of mass acceleration?

  $\vec{F}_{net} =$$M\vec{a}_{CM}$
• When applying Newton's Second Law to a system of particles, we can treat the entire system as a single point object located at the center of mass.
• Applying Newton's Second Law to the system's center of mass simplifies the analysis by treating the system as a single point object.

  True
• For a system of three particles with masses 2 kg, 3 kg, and 4 kg, if the net force on the system is (10 N, 5 N), the acceleration of the center of mass is (1 m / s^{2}, 0.5 m / s^{2}).
• What is the center of mass (CM) of a single object?

  Geometric center
• What is the CM position for a rod of length $L$ with a linear mass density $\lambda(x) =$$ax$?


  $\frac{2}{3}L$
• Defining a system in physics helps identify its boundaries and interactions.


  True
• What formula is used to calculate the center of mass for a system of particles?

  $\vec{r}_{CM} =$$\frac{\sum m_{i} \vec{r}_{i}}{\sum m_{i}}$
• Why is knowing the center of mass important?

  Treat system as a point object
• External forces on a system are considered outside the system boundaries.


  True
• Which laws of physics depend on understanding the concept of a system?

  Newton's laws of motion
• What does defining a system simplify in physics?

  Applying physical principles
• The center of mass for a system of particles is calculated using a weighted average of their masses and positions.
• What is the next step in calculating the center of mass for a discrete mass distribution after defining the system?

  Determining particle positions
• In the formula for the center of mass, $m_{i}$ represents the mass of the i-th particle
• When analyzing a system of particles, the system is treated as a single object
• Match the variables in the center of mass formula with their definitions:

  $\vec{r}_{CM}$ ↔️ Position vector of CM

  $m_{i}$ ↔️ Mass of the i-th particle

  $\vec{r}_{i}$ ↔️ Position vector of the i-th particle
• What is a continuous mass distribution?

  Mass distributed throughout
• What is a system in physics?

  Object under study
• What is the formula for calculating the position vector of the center of mass for a system of particles?

  $\vec{r}_{CM} =$$\frac{\sum m_{i} \vec{r}_{i}}{\sum m_{i}}$
• For a single object, the center of mass is always its geometric center.


  True
• For a discrete mass distribution, the center of mass is calculated using the summation formula.