2.1 Systems and Center of Mass

• A system in physics is a defined portion of the universe
• The center of mass is the point where the total mass of an object is concentrated
• Match the concept with its definition:

  System ↔️ A defined portion of the universe for analysis

  Center of Mass ↔️ The point where an object's total mass is concentrated
• The center of mass simplifies the analysis of forces and motion.
• The interaction with the surroundings involves the exchange of energy and momentum
• Steps to calculate the center of mass for a discrete system

  1️⃣ Identify the mass and position of each particle

  2️⃣ Apply the formula to calculate the center of mass
• The position vector of the center of mass is denoted by \vec{R}
• What is the center of mass for a system with two particles, \(m_1 = 2\) kg at \(\vec{r_1} = (1, 1)\) m and \(m_2 = 3\) kg at \(\vec{r_2} = (4, 5)\) m?

  (2.8, 3.4) m
• The formula for the center of mass of continuous systems involves integration.
• The mass density function in the formula for continuous systems is denoted by \rho(\vec{r})
• For continuous systems, the center of mass is calculated using a different formula
• \(\vec{R}\) represents the position vector of the center of mass in the continuous systems formula.
• \(\rho(\vec{r})\) is the mass density function, which gives the mass per unit volume
• \(dV\) in the continuous systems formula represents an infinitesimal volume element
• Steps to apply the continuous systems formula for calculating the center of mass

  1️⃣ Determine the mass density function \(\rho(\vec{r})\)

  2️⃣ Integrate the numerator and denominator over the entire volume of the system
• In physics, a system is a defined portion of the universe isolated for study
• Match the concept with its definition:

  System ↔️ A defined portion of the universe for analysis

  Center of Mass ↔️ The point where an object's total mass is concentrated
• The center of mass simplifies the analysis of motion and forces by treating all mass as if it were concentrated at a single point
• \(\vec{R}\) in the discrete systems formula represents the position vector of the center of mass
• \(\vec{r_{i}}\) in the discrete systems formula is the position vector of each particle
• What does the variable $\vec{R}$ represent in the formula for the center of mass of a discrete system?


  Position vector of the center of mass
• The formula for calculating the center of mass of a discrete system is \vec{R} = $\frac{\sum m_{i} \vec{r_{i}}}{\sum m_{i}}$
• What does the variable $m_{i}$ represent in the formula for the center of mass of a discrete system?


  Mass of each particle
• The variable $\vec{r_{i}}$ in the discrete system formula represents the position vector of each particle.
• Match the characteristics with the type of system:

  Continuous Systems ↔️ Continuous mass distribution

  Discrete Systems ↔️ Discrete particles
• What is $\rho(\vec{r})$ in the formula for the center of mass of a continuous system?


  Mass density function
• The formula for the center of mass of a continuous system is \vec{R} = $\frac{\int \rho(\vec{r}) \vec{r} dV}{\int \rho(\vec{r}) dV}$
• The center of mass for a uniform solid cube with side length \(a\) is at $(\frac{a}{2}, \frac{a}{2}, \frac{a}{2})$.
• What type of forces govern the motion of the center of mass of a system?

  External forces
• Steps to calculate the center of mass of a discrete system:

  1️⃣ Identify the mass and position of each particle

  2️⃣ Apply the formula $\vec{R} =$$\frac{\sum m_{i} \vec{r_{i}}}{\sum m_{i}}$

  3️⃣ Calculate the numerator

  4️⃣ Calculate the denominator

  5️⃣ Divide numerator by denominator
• The center of mass for a uniform solid cube is at its geometric center.