# 3.4: Notation

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In this section I am going to suppose that we \( n\) particles scattered through three-dimensional space. We shall be deriving some general properties and theorems – and, to the extent that a solid body can be considered to be made up of a system of particles, these properties and theorems will apply equally to a solid body.

In the Figure III.5, I have drawn just two of the particles, (the rest of them are left to your imagination) and the centre of mass C of the system.

A given particle may have an *external force *\( \bf{F}_{i}\) acting upon it. (It may, of course, have *several* external forces acting on it, but I mean by \( \bf{F}_{i}\) the vector sum of all the external forces acting on the *i *th particle.) It may also interact with the other particles in the system, and consequently it may have *internal forces* \( \bf{F}_{ij}\) acting upon it, where \( j\) goes from 1 to \( n\) except for \( i\). I define the vector sum \( \bf F = \sum F_{i}\) as the total external force acting upon the *system*.

I am going to establish the following notation for the purposes of this chapter.

- Mass of the
*\( i\)*th particle = \( m_{i}\) - Total mass of the system \(M= \sum m_{i}\)
- Position vector of the
*\( i\)*th particle referred to a fixed point O: \( \textbf{r}_{i} = x_{i} \hat{\textbf{x}} + y_{i} \hat{\textbf{y}} + z_{i} \hat{\textbf{z}}\) - Velocity of the
*\( i\)*th particle referred to a fixed point O: \( \textbf{r}_{i}\) or \( \textbf{v}_{i}\) (Speed = \( v_{i}\)) - Linear momentum of the
*\( i\)*th particle referred to a fixed point O: \(\textbf{p}_{i} = m_{i} \textbf{v}_{i}\) - Linear momentum of the
*system*: \(\textbf{P} = \sum \textbf{P}_{i} = \sum m_{i} \textbf{v}_{i}\) - External force on the
*\( i\)*th particle: \(\textbf{F}_{i}\) - Total external force on the system: \( \textbf{F} = \sum \textbf{F}_{i}\)
- Angular momentum (moment of momentum) of the
*\( i\)*th particle referred to a fixed point O: \[\textbf{l}_{i} = \textbf{r}_{i} \times \textbf{p}_{i}\] - Angular momentum of the system: \( \textbf{L} = \sum \textbf{l}_{i} = \sum \textbf{r}_{i} \times \textbf{p}_{i}\)
- Torque on the
*i*th particle referred to a fixed point O: \( \boldsymbol\tau_{i} = \textbf{r}_{i} \times \textbf{F}_{i}\) - Total external torque on the system with respect to the origin: \[ \boldsymbol\tau = \sum \boldsymbol\tau_{i} = \sum \textbf{r}_{i} \times \textbf{F}_{i}\]

Kinetic energy of the system: (We are dealing with a system of *particles *– so we are dealing only with *translational *kinetic energy – no rotation or vibration):

\( T = \sum \frac{1}{2} m_{i}v_{i}^{2} \)

Position vector of the *centre of mass *referred to a fixed point O: \(\overline{\textbf{r}}_{i} =\overline{x}\hat{\textbf{x}} + \overline{y}\hat{\textbf{y}} + \overline{z}\hat{\textbf{z}} \)

The centre of mass is defined by \(M \overline{\textbf{r}} = \sum m_{i} \textbf{r}_{i}\)

Velocity of the centre of mass referred to a fixed point O: \( \overline{\textbf{r}} = \overline{\textbf{v}}\) (Speed = \( \overline{v}\) )

For position vectors, unprimed single-subscript symbols will refer to O. Primed single-subscript symbols will refer to C. This will be clear, I hope, from Figure III.5.

Position vector of the *\( i\) *th particle *referred to the centre of mass* C: \( \textbf{r}'_{i} = \textbf{r}_{i} - \overline{\textbf{r}}_{i}\)

Position vector of particle *\( j\)* with respect to particle *\( i\)*: \(\textbf{r}_{ij} = \textbf{r}_{j} - \textbf{r}_{i}\)

(Internal) force exerted on particle *\( i\)* by particle *\( j\)*: \(\textbf{F}_{ij}\)

(Internal) force exerted on particle *\( j\)* by particle *\( i\)*: \(\textbf{F}_{ji}\)

If the force between two particles is *repulsive *(e.g. between electrically-charged particles of the same sign), then \(\textbf{F}_{ji}\) and \(\textbf{r}_{ji}\) are in the same direction. But if the force is an *attractive* force, \(\textbf{F}_{ji}\) and \(\textbf{r}_{ji}\) are in opposite directions.

According to Newton’s Third Law of Motion (Lex III), \(\textbf{F}_{ij} = -\textbf{F}_{ji} \)

Total angular momentum of system referred to the centre of mass C: \(\textbf{L}_{C}\)

Total external torque on system referred to the centre of mass C: \(\boldsymbol \tau_{C}\)

For the velocity of the centre of mass I may use either \( \dot{\overline{\textbf{r}}}\) or \( \overline{\textbf{v}}\)

O is an arbitrary origin of coordinates. C is the centre of mass.

Note that

\begin{equation} \ \textbf{r}_{i} = \overline{\textbf{r}} + \textbf{r}_{i}^{\prime} \tag{3.4.1}\label{eq:3.4.1} \end{equation}

and therefore

\begin{equation}\ \dot{\overline{\textbf{r}}_{i}} = \dot{\overline{\textbf{r}}} + \dot{\textbf{r}_{i}^{\prime}}; \tag{3.4.2}\label{eq:3.4.2} \end{equation}

that is to say

\begin{equation} \ \textbf{v}_{i} = \overline{\textbf{v}} + \textbf{v}_{i}^{\prime} \tag{3.4.3}\label{eq:3.4.3} \end{equation}

Note also that

\begin{equation} \ \sum m_{i} \textbf{r}_{i}^{\prime} = 0 \tag{3.4.4}\label{eq:3.4.4} \end{equation}

Note further that

\begin{equation} \sum m_{i} \textbf{v}'_i= \sum m_{i}(\textbf{v}_i - \overline{\textbf{v}}) = \sum m_{i}\textbf{v}_{i} -\overline{\textbf{v}} \sum m_{i} = M\overline{\textbf{v}} - \overline{\textbf{v}}M = 0 \tag{3.4.5}\label{eq:3.4.5} \end{equation}

That is, *the total linear momentum with respect to the centre of mass is zero.*

Having established our notation, we now move on to some theorems concerning systems of particles. It may be more useful for you to conjure up a physical picture in your mind what the following theorems mean than to memorize the details of the derivations.