Point Particle Model and Center of Mass

The center of mass (COM) is the point where we can treat an object or system as if all its mass were concentrated there.

For a system of discrete particles, the COM position is found using a weighted average. In one dimension:

$$x_{\text{cm}} = \frac{\sum m_i x_i}{\sum m_i} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + \cdots}{m_1 + m_2 + m_3 + \cdots}$$

Here $m_i$ is the mass of each particle and $x_i$ is its position. Heavier masses contribute more to the COM location.

The same formula applies in the $y$-direction:

$y_{\text{cm}} = \dfrac{\sum m_i y_i}{\sum m_i}$.

When working with a few discrete masses in 1D, apply the above formula in either the $x$ or $y$ direction to find the point that represents where the entire system's mass can be cons

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