What is Center of Mass?
The center of mass is a unique point in a system of objects where the weighted average of all parts of the system's mass is located. It's the point where the entire mass of the system can be considered to be concentrated for the purpose of analyzing its translational motion. In simpler terms, it is the balance point of an object or system.
Formula for the Center of Mass
For a system of 'n' discrete point masses ($m_1, m_2, ..., m_n$) located at coordinates ($(x_1, y_1), (x_2, y_2), ..., (x_n, y_n)$), the coordinates of the center of mass ($X_{CM}, Y_{CM}$) are calculated using the following formulas:
X-coordinate of Center of Mass ($X_{CM}$):
$$X_{CM}=\frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i}=\frac{m_1x_1 + m_2x_2 + \dots + m_nx_n}{m_1 + m_2 + \dots + m_n}$$
Y-coordinate of Center of Mass ($Y_{CM}$):
$$Y_{CM}=\frac{\sum_{i=1}^{n} m_i y_i}{\sum_{i=1}^{n} m_i}=\frac{m_1y_1 + m_2y_2 + \dots + m_ny_n}{m_1 + m_2 + \dots + m_n}$$
Where:
- $m_i$ is the mass of the i-th object.
- $x_i$ is the x-coordinate of the i-th object.
- $y_i$ is the y-coordinate of the i-th object.
How to Use This Calculator
- Enter the mass and the X and Y coordinates for at least two objects in the provided fields.
- If you have more than two objects, click the "Add Another Mass" button to create new input fields.
- The Center of Mass coordinates (X and Y) are calculated automatically and displayed in the green-highlighted result boxes.
- Any change in the input values will instantly update the result.
- Click the "Clear" button to reset all fields and start a new calculation.
