# System of Particles Notes

Here is a collection of notes on the dynamics of a system of particles. The main definitions and relationships are summarized below.

*Definition of center of mass* \( \vec r_G \) (derivation here):

$$ \boldsymbol{\vec r}_G = \frac{\sum_{j=1}^N m_j \, \boldsymbol{\vec r}_j}{\sum_{j=1}^N m_j} = \frac{\sum m_j \, \boldsymbol{\vec r}_j}{m_{tot}}$$

*Newton’s Second Law for systems of particles* (derivation here):

$$ \sum \boldsymbol{\vec F}^{\,(ext)} = m_{tot} \, \boldsymbol{\vec a}_G.$$

*Moment equation for a system of particles* (derivation here):

$$ \sum \boldsymbol{\vec M}_p^{\,(ext)} = \boldsymbol{\dot H}^{(tot)}_p.$$

#### Definition of Center of Mass

In your statics course, you probably discussed something called “center of gravity”. Now were going to discuss something very similar called the center of mass. This set of notes provides a review.

Here is an interactive tool that displays the center of mass of a collection of three objects. Click and drag the red, blue and green objects. Adjust the masses with the sliders. If you want to see the full size version, CLICK HERE.

#### Newton’s Second Law for a System of Particles

Here is a set of notes in which we start with Newton’s Second Law for single particles and develops a theory for collections of particles.

#### Moment Equation for a System of Particles

In this set of notes we take moments for each particle in the collection, add them up, and see what happens.