- #1

Portuga

- 56

- 6

## Homework Statement

Hello gentlemen!

I came across a real challeging problem about static, which states

the following: assume a force system is equivalent to a force [itex]\vec{F}_{1}[/itex] and a couple [itex]M_{1}\vec{k}[/itex] acting at a point [itex]\vec{r}_{1}[/itex]. Find some point [itex]\vec{r}_{2}[/itex] and a force [itex]\vec{F}_{2}[/itex] so that [itex]\vec{F}_{2}[/itex] acting at [itex]\vec{r}_{2}[/itex] is equivalent to [itex]\vec{M}_{1}[/itex] acting at [itex]\vec{r}_{1}[/itex].

As it is a little more difficult problem, the author provided a solution:

[tex]

\vec{r}_{2}=\vec{r}_{1}+\frac{\vec{F}_{1}\times\hat{k}M_{1}}{F_{1}^{2}},

[/tex] and

[tex]

\vec{F}_{2}=\vec{F}_{1}.

[/tex]

## Homework Equations

My first attempt was to use the equivalence of the couples:

[tex]

\vec{r}_{1}\times\vec{F}_{1}=\vec{r}_{2}\times\vec{F}_{2}=M_{1}\hat{k},

[/tex]

and the BAC - CAB rule for double cross product: [itex] \vec A \times \vec B \times \vec C = \vec B (\vec A \ldotp \vec C) - \vec C (\vec A \ldotp \vec B) [/itex]

## The Attempt at a Solution

That [itex]\vec{F}_1 = \vec{F}_2[/itex], it's obvious, because of the equivalence of the forces systems. So I focused on the equivalence of the couples.

I realized that only the last two members of the vectorial equation were interesting for this:

[tex]

\begin{aligned} & \vec{F}_{1}\times\vec{r}_{2}\times\vec{F}_{2}=\vec{F}_{1}\times M_{1}\hat{k}\\

\Rightarrow & \vec{r}_{2}\left(\vec{F}_{1}\ldotp\vec{F}_{2}\right)-\left(\vec{F}_{1}\ldotp\vec{r}_{2}\right)\vec{F}_{2}=\vec{F}_{1}\times M_{1}\hat{k}\\

\Rightarrow & \vec{r}_{2}=\frac{\left(\vec{F}_{1}\ldotp\vec{r}_{2}\right)\vec{F}_{2}+\vec{F}_{1}\times M_{1}\hat{k}}{\vec{F}_{1}\ldotp\vec{F}_{2}}.

\end{aligned}

[/tex]So my point is: if there is freedom to choose both [itex]\vec{r}_{2}[/itex] and [itex]\vec{F}_{2}[/itex] , it's not possible to assume that these vectors should be related to the parameters of the problem, [itex]\vec{r}_{1}[/itex], [itex]\vec{F}_{1}[/itex] and [itex]M_{1}\hat{k}[/itex] as the author requires in his solution. Am I right or am I missing something really important on this issue?