> [!summary] The center of mass is a point where you can imagine all the mass of a system without changing the motion of a body. The distance covered from a reference point is the center of mass displacement. > **Key Equations:** > Center of mass displacement: $\vec{r_{cm}} = \frac{\displaystyle \sum_{i}m_{i}\vec{r_{i}}}{\displaystyle \sum_{i} m_{i}}$ > Verification of center of mass displacement: $\sum_{i} m_{i} r_{cm_{i}} =0$ >[!info]+ Read Time **⏱ 2 mins** # Definition The center of mass of a system of particles or a rigid body is a point where you can imagine the entire mass of a system without changing the motion of the body. It's useful in finding a balance point. Or a point where all the position vectors on a rigid body or system of particles disappear. Mathematically, the displacement of the center of mass is found using a reference point which can be chosen anywhere on an axis. Which means the displacement of the center of mass depends on where you start the displacement. The displacement of the center of mass is the sum of position vectors from the reference point to each mass element over the total mass. $ \vec{r_{cm}} = \frac{\displaystyle \sum_{i}m_{i}\vec{r_{i}}}{\displaystyle \sum_{i} m_{i}} $ ![[com_1.png|400]] > [!note] Explanation Example of a center of mass calculation system of particles. Add the total mass elements and their displacement vectors over the total mass, which gives you the displacement vector of the center of mass If, instead, you wanted to verify a point as the center of mass. Mathematically, this can be done by taking the sum of all mass elements with reference to the center of mass point. The result should be zero. $ \sum_{i} m_{i} r_{cm_{i}} =0 $ # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/ol1COj0LACs?si=k92kiyZCenmdh4Qs" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>