Section 9.2 The Kinetic Screw of Material System
Recall from
Section 4.8 that, given a vector field
\(P\in \Sigma \mapsto {\bf u}_P\) defined over a material system
\(\Sigma\text{,}\) the vector field defined by
\begin{equation*}
A \mapsto \int_{\Si} \br_{AP} \times {\bf u}_P \, dm.
\end{equation*}
defines a screw whose resultant is \(\int_{\Si} {\bf u}_P \, dm\text{.}\) We can define two screws in this manner by choosing first the velocity then the acceleration fields of \(\Sigma\) relative to a referential \(\cE\text{.}\) Keep in mind that system \(\Si\) is not necessarily a rigid body. Thus, field \({\bf u}_P\) need not define a screw.
The first such screw is obtained by choosing \({\bf u}_P =\vel_{P/\cE}\text{.}\)
Definition 9.2.1. Kinetic Screw of a Material System.
The kinetic screw of material system \(\Si\) is the screw denoted \(\{ \cH _{\Si / \cE } \}\) corresponding to the vector field \(A \mapsto \bH_{A, \Si / \cE} = \int_{\Si} \br_{AP} \times \vel_{P/\cE} \, dm\text{.}\) Its resultant, called linear momentum of \(\Sigma\) (relative to \(\cE\)), is given by
\begin{equation}
\bL_{\Si / \cE} = \int_{\Si} \vel_{P/\cE} \, dm.\tag{9.2.1}
\end{equation}
Moment \(\bH_{A, \Si / \cE}\) is called angular momentum of system \(\Si\) relative to \(\cE\) about point \(A\text{.}\)
We may denote the angular momentum (also known as the moment of momentum) about point \(A\) simply as \(\bH_{A, \Si}\text{,}\) or even \(\bH_A\) when the context permits it.
Theorem 9.2.2. Theorem of the Mass Center.
The linear momentum (relative to referential \(\cE\text{.}\) of a material system \(\Si\) of mass \(m\) and mass center \(G\) is equal to the linear momentum of a fictitious particle of mass \(m\) coinciding with \(G\)
\begin{equation}
\int_{\Si} \vel_{P/\cE} \, dm = m \vel_{G/\cE} \tag{9.2.2}
\end{equation}
Hence, the kinetic screw of \(\Sigma\) can be written in the following form
\begin{equation}
\{ \cH_{\Si/ \cE} \} =
\begin{Bmatrix}
m \vel_{G/\cE} \\
\bH_{A}
\end{Bmatrix}\tag{9.2.3}
\end{equation}
and the screw property of the angular momentum can be stated in the form
\begin{equation}
\bH_{A} = \bH_{B} + {\bf r}_{AB} \times m \vel_{G/\cE}\tag{9.2.4}
\end{equation}
between any two points \(A\) and \(B\text{.}\)
Proof.
By definition, we have for any point
\(O\) fixed in
\(\cE\text{,}\) \(m \br_{OG} = \int_{\Si} \br_{OP} \, dm\text{,}\) and upon taking the time derivative in
\(\cE\) (using
(9.1.1)), we obtain
\begin{equation*}
\left({d \over dt} \int_{\Si} \br_{OP} \, dm \right)_\cE =
\left( {d \over dt} m \br_{OG} \right)_{\cE} = m \vel_{G/\cE}
\end{equation*}
recalling that the mass of system \(\Si\) is constant.
Hence, we have at hand an easy way of determining the linear momentum of a system \(\Si\) from the kinematics of a single particle located at the mass center of system \(\Si\text{.}\) The angular momentum of \(\Si\) is more difficult to determine for arbitrary material systems. However, for a rigid body \(\cB\text{,}\) its determination is straightforward thanks to the fact that the velocity field \(P\in \cB\mapsto \vel_{P/\cE}\) defines a screw. Before we devote our attention to its calculation, we introduce another screw which will play a fundamental role in dynamics.
