Center of Mass Calculator

What Is the Center of Mass?

The center of mass (often abbreviated as CM or COM) of a distribution of mass in space is the unique point where the weighted relative position of the distributed mass sums to zero. In classical physics and mechanics, when an external net force acts upon an extended body or system of particles, the system translates as if its entire mass were concentrated at this single point.

In a uniform gravitational field, the center of mass coincides exactly with the center of gravity. Finding the center of mass is critical in structural engineering, aerospace trajectory planning, vehicle stability, sports biomechanics, and robotics.

Center of Mass Formulas for Discrete Particles

For a discrete system containing $N$ point masses $m_1, m_2, \dots, m_N$ positioned at coordinates $\vec{r}_i = (x_i, y_i, z_i)$, the total mass $M$ of the system is:

$$M = \sum_{i=1}^{N} m_i = m_1 + m_2 + \dots + m_N$$

The coordinates of the center of mass $(X_{cm}, Y_{cm}, Z_{cm})$ are calculated by dividing the first moments of mass by the total mass:

$$X_{cm} = \frac{\sum_{i=1}^{N} m_i x_i}{M} = \frac{m_1 x_1 + m_2 x_2 + \dots + m_N x_N}{m_1 + m_2 + \dots + m_N}$$

$$Y_{cm} = \frac{\sum_{i=1}^{N} m_i y_i}{M} = \frac{m_1 y_1 + m_2 y_2 + \dots + m_N y_N}{m_1 + m_2 + \dots + m_N}$$

$$Z_{cm} = \frac{\sum_{i=1}^{N} m_i z_i}{M} = \frac{m_1 z_1 + m_2 z_2 + \dots + m_N z_N}{m_1 + m_2 + \dots + m_N}$$

Step-by-Step Calculation Example

Consider three objects on a 2D plane:

• Object 1: Mass $m_1 = 2\text{ kg}$ at $(0, 0)$
• Object 2: Mass $m_2 = 3\text{ kg}$ at $(4, 0)$
• Object 3: Mass $m_3 = 5\text{ kg}$ at $(2, 4)$

Step 1: Total mass: $M = 2 + 3 + 5 = 10\text{ kg}$.
Step 2: Moment in $X$: $M_x = (2 \cdot 0) + (3 \cdot 4) + (5 \cdot 2) = 0 + 12 + 10 = 22\text{ kg}\cdot\text{m}$.
Step 3: Moment in $Y$: $M_y = (2 \cdot 0) + (3 \cdot 0) + (5 \cdot 4) = 0 + 0 + 20 = 20\text{ kg}\cdot\text{m}$.
Step 4: Center of mass coordinates:
$$X_{cm} = \frac{22}{10} = 2.2\text{ m}, \quad Y_{cm} = \frac{20}{10} = 2.0\text{ m}$$ The center of mass is located at $(2.2, 2.0)$.

Difference Between Center of Mass and Centroid

While closely related, the two concepts have a distinct distinction:

• Centroid (Geometric Center): The arithmetic mean position of all points in a shape, assuming completely uniform density throughout.
• Center of Mass: Takes into account the physical mass distribution of the object. If some parts are heavier than others, the center of mass shifts toward the heavier sections.

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