Case 2: if ${\bf r}_{A_1 A_2} \neq {\bf 0}$, then the three vectors ${\bf S}_1$, ${\bf S}_2$, and ${\bf r}_{A_1 A_2}$ are coplanar, or two of these vectors are collinear. In the first case, the sliders' axes are intersecting or parallel.
In the second case, either ${\bf S}_1$ and ${\bf S}_2$ are collinear or ${\bf r}_{A_1 A_2}$ and ${\bf S}_1$ (or ${\bf S}_2$) are collinear: again, this implies that the sliders' axes are either intersecting or parallel.
In conclusion, if the sum of two sliders is a slider, then their axes are either intersecting or parallel.
The converse is true if ${\bf S}_1 + {\bf S}_2\neq {\bf 0}$.
Sunday, March 1, 2015
SUM OF TWO SLIDERS
Consider two sliders $\{ {\cal S}_1 \}$ and $\{ {\cal S}_2 \}$ defined by
\[
\{ {\cal S}_1 \} = \begin{Bmatrix}{\bf S}_1 \\ {\bf 0} \end{Bmatrix}_{A_1}, \quad
\{ {\cal S}_2 \} = \begin{Bmatrix} {\bf S}_2 \\ {\bf 0} \end{Bmatrix}_{A_2}, \quad
\]
Assuming ${\bf S}_1 + {\bf S}_2\neq {\bf 0}$, let's find the conditions for which the sum
$\{ {\cal S}_1 \}+ \{ {\cal S}_2 \}$ is a slider. A necessary and sufficient condition is
that $\{ {\cal S}_1 \}\cdot \{ {\cal S}_2 \} =0$ which can be stated as
\[
{\bf S}_1 \cdot ( {\bf S}_2 \times {\bf r}_{A_1 A_2} ) = 0
\]
Case 1: if ${\bf r}_{A_1 A_2} ={\bf 0}$, then the sliders' axes are intersecting.
