Centre of Mass

The centre of mass in one dimension is given by $x_{cm}=\frac{{\sum }_{i=1}^n{m}_i{x}_i}{M_T}$

The centre of mass in more than one dimension is given by ${\vec{r}}_{cm}=\frac{{\sum }_{i=1}^n{m}_i{\vec{r}}_i}{{\sum }_{i=1}^n{m}_i}$

For an object with a continuous mass distribution, the centre of mass is given by ${\vec{r}}_{cm}=\frac{{\int }_V\vec{r}dm}{{\int }_{V}dm}$ The above seems complicated, but really it is $x_{CM}=\frac{1}{M}\int xdm$ $y_{CM}=\frac{1}{M}\int ydm$ $z_{CM}=\frac{1}{M}\int zdm$

Centroid

• The centroid doesn’t necessarily have to lie on the body
• It will lie on an axis of symmetry
• For homogeneous bodies, the centroid is the same as the center of mass

#gap-in-knowledge https://www.youtube.com/watch?v=SMy8DD2OU8M&ab_channel=PhysicsNinja

Related

• Used in Linear Momentum