Lecture Notes: Calculating the Center of Mass

Key Concepts

• Center of Mass: The point where the total mass of a system can be considered to be concentrated. It can be found by considering the mass distribution and their positions.
• Mass Distribution on a Plane: In this problem, we are given point masses distributed on a two-dimensional plane.

Steps to Calculate Center of Mass

1. Identify Masses and Positions:

   • You are given the masses and their coordinates on a plane.
   • Assign these masses as M1, M2, M3, M4 based on your preference; the choice of labeling does not affect the final result.

2. Equation for Center of Mass (X-coordinate):

   • The formula for the x-coordinate of the center of mass is:

     [ X_{cm} = \frac{M_1X_1 + M_2X_2 + M_3X_3 + M_4X_4}{M_1 + M_2 + M_3 + M_4} ]

3. Plug in Values for X-coordinate:

   • Example values:
     • M1 = 2 kg with x-coordinate = 0.2 m
     • M2 = 6 kg with x-coordinate = 5 m
     • M3 = 4 kg with x-coordinate = 3 m
     • M4 = 8 kg with x-coordinate = 0 m
   • Total mass = 2 + 6 + 4 + 8 = 20 kg
   • Result for x-coordinate: 2.3 meters

4. Equation for Center of Mass (Y-coordinate):

   • The formula for the y-coordinate of the center of mass is:

     [ Y_{cm} = \frac{M_1Y_1 + M_2Y_2 + M_3Y_3 + M_4Y_4}{M_1 + M_2 + M_3 + M_4} ]

5. Plug in Values for Y-coordinate:

   • Example values:
     • M1 = 2 kg with y-coordinate from origin
     • M2 = 6 kg with specific y-coordinate
     • M3 = 4 kg with y-coordinate = 5 m
     • M4 = 8 kg with y-coordinate = 3 m
   • Total mass = 20 kg
   • Result for y-coordinate: 2.8 meters

Conclusion

• Final Center of Mass Coordinates:

  • ( X_{cm} = 2.3 ) meters
  • ( Y_{cm} = 2.8 ) meters

• The center of mass coordinates (X, Y) illustrate the weighted average position of the point masses considering their mass and location on the plane.