In Sect. 6.3 we determined that the overall translational equation
of motion of a general -component system can
be written in the form

(457) |

Equation (456) effectively determines the *translational motion* of the system's centre of mass.
Note, however, that in order to fully determine the motion of the system we must also follow its
*rotational motion* about its centre of mass (or any other convenient reference point).
In Sect. 9.4 we determined that the overall rotational equation of motion
of a general -component system (with central internal forces) can
be written in the form

(459) |

What conditions must be satisfied by the various external forces and torques acting
on the system if it is to remain stationary in time? Well,
if the system does not evolve in time then its net linear momentum, ,
and its net angular momentum, , must both remain constant.
In other words,
.
It
follows from Eqs. (456) and (458) that

(460) | |||

(461) |

In other words, the net external force acting on system must be zero, and the net external torque acting on the system must be zero. To be more exact:

andThe components of the net external force acting along any three independent directions must all be zero;

In a nutshell, these are the principles of statics.The magnitudes of the net external torques acting about any three independent axes (passing through the origin of the coordinate system) must all be zero.

It is clear that the above principles are *necessary* conditions for a general physical
system not to evolve in time. But, are they also *sufficient* conditions? In other
words, is it necessarily true that a general system which satisfies these conditions does not
exhibit any time variation? The answer to this question is as follows: if the
system under investigation is a *rigid body*, such that the motion of any
component of the body necessarily implies the motion of the whole body, then the
above principles are necessary and sufficient conditions for the existence of an equilibrium
state. On the other hand, if the system is not a rigid body, so
that some components of the body can move independently of others, then
the above conditions only guarantee that the system remains static in an average
sense.

Before we attempt to apply the principles of statics, there are a couple
of important points which need clarification. Firstly, does it matter about
which point we calculate the net torque acting on the system? To be more
exact, if we determine that the net torque acting about a given point
is zero does this necessarily imply that the net torque acting about any
other point is also zero? Well,

(462) |

(463) |

(464) |

(465) |

Another question which needs clarification is as follows. At which point should
we assume that the weight of the system acts in order to calculate the contribution
of the weight to the net torque acting about a given point? Actually, in Sect. 8.11,
we effectively answered this question by assuming that the weight
acts at the centre of mass of the system. Let
us now justify this assumption. The external force acting on the th component
of the system due to its weight is

(466) |

(467) |