2.1 Systems and Center of Mass

• In physics, a system refers to the object or set of objects being analyzed
• Why is defining a system crucial in physics?

  Identifies boundaries and interactions
• The center of mass is the average position of all the mass in the system, weighted by the mass of each individual part
• The center of mass simplifies the analysis of a system's motion
• What is the formula for $y_{cm}$ in a two-dimensional system?


  y_{cm} = \frac{\sum m_{i} y_{i}}{\sum m_{i}}</latex>
• To calculate the center of mass, the first step is to multiply the mass of each particle by its x-coordinate.


  True
• For two particles with masses $m_{1} =$$2\text{ kg}$ at x_{1} = 3\text{ m}</latex> and $m_{2} =$$3\text{ kg}$ at $x_{2} =$$8\text{ m}$, the x-coordinate of the center of mass is 6 meters.
• What is the formula for calculating the x-coordinate of the center of mass for discrete particles?

  x_{cm} = \frac{\sum m_{i} x_{i}}{\sum m_{i}}</latex>
• The formula for the x-coordinate of the center of mass for continuous objects is $x_{cm} =$$\frac{\int x \, dm}{\int dm}$.


  True
• What is the x-coordinate of the center of mass for a uniform rod of mass M and length L?

  $x_{cm} =$$\frac{L}{2}$
• What are the edges that separate a system from its environment called?

  Boundaries
• Why is the center of mass an important concept in physics?

  Simplifies motion analysis
• The center of mass for discrete particles is calculated using integrals.

  False
• How is the center of mass calculated for continuous objects?

  Using integrals
• The center of mass simplifies the analysis of translational dynamics.

  True
• For continuous objects, $dm$ can be expressed in terms of volume or area
• Match the integral type with its formula:

  Volume ↔️ $x_{cm} =$$\frac{\int x \, \rho \, dV}{\int \rho \, dV}$

  Surface ↔️ $x_{cm} =$$\frac{\int x \, \rho \, dA}{\int \rho \, dA}$
• What is the center of mass for a uniform rod of mass M and length L?

  $L / 2$
• Internal forces affect the motion of the center of mass.

  False
• The center of mass is the point where the total mass of an object is concentrated.


  True
• What does $x_{cm}$ represent in the center of mass formula?


  X-coordinate of the center of mass
• Steps to calculate the center of mass for discrete particles

  1️⃣ Multiply the mass of each particle by its x-coordinate

  2️⃣ Sum these values

  3️⃣ Divide by the total mass of all particles
• What is the formula for calculating the x-coordinate of the center of mass for discrete particles?

  $x_{cm} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$
• What additional formula is needed for a two-dimensional system to calculate the center of mass?

  $y_{cm} =$$\frac{\sum m_{i} y_{i}}{\sum m_{i}}$
• Match the dimension with the corresponding formula for the center of mass:

  One Dimension ↔️ $x_{cm} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$

  Two Dimensions ↔️ $x_{cm} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$ and $y_{cm} =$$\frac{\sum m_{i} y_{i}}{\sum m_{i}}$
• What mathematical tool is used to calculate the center of mass for continuous objects?

  Integrals
• Match the aspect with the correct type of object:

  Formula ↔️ $x_{cm} =$$\frac{\int x \, dm}{\int dm}$ for continuous objects

  Calculation ↔️ Summations for discrete particles

  Mass Distribution ↔️ Continuous for continuous objects
• Defining a system in physics allows us to identify the boundaries, internal interactions, and external interactions.


  True
• For continuous objects, the center of mass is calculated using integrals.

  True
• The center of mass is the point where the total mass of an object or system is concentrated
• For discrete particles, the x-coordinate of the center of mass is given by the formula $x_{cm} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$, where $m_{i}$ is the mass of the i-th particle and $x_{i}$ is its coordinate
• For continuous objects, the x-coordinate of the center of mass is given by the formula $x_{cm} =$$\frac{\int x \, dm}{\int dm}$, where $dm$ is the infinitesimal mass element
• Steps to calculate the center of mass for two discrete particles

  1️⃣ Identify the mass and position of each particle

  2️⃣ Apply the formula $x_{cm} =$$\frac{\sum m_{i} x_{i}}{\sum m_{i}}$

  3️⃣ Sum the products of mass and position

  4️⃣ Divide by the total mass

  5️⃣ The result is the x-coordinate of the center of mass
• What does $\rho$ represent in the formulas for continuous objects?


  Density
• If an object has uniform density, $\rho$ can be taken out of the integral.


  True
• The motion of the center of mass is governed by Newton's Laws, specifically the second law
• What is the formula relating external force, mass, and acceleration of the center of mass?

  F_{ext} = M a_{cm}</latex>