# General equivalent couple - force system

• Portuga
In summary: So, the result would be that r1 would increase and become larger than r2. If that is what you were looking for, then I think you are incorrect. There must be a way to relate the vectors without changing the magnitude of r1.I don't think so either. What am I missing?I think you are missing the equivalence of the couples.
Portuga

## 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 $\vec{F}_{1}$ and a couple $M_{1}\vec{k}$ acting at a point $\vec{r}_{1}$. Find some point $\vec{r}_{2}$ and a force $\vec{F}_{2}$ so that $\vec{F}_{2}$ acting at $\vec{r}_{2}$ is equivalent to $\vec{M}_{1}$ acting at $\vec{r}_{1}$.

As it is a little more difficult problem, the author provided a solution:
$$\vec{r}_{2}=\vec{r}_{1}+\frac{\vec{F}_{1}\times\hat{k}M_{1}}{F_{1}^{2}},$$ and
$$\vec{F}_{2}=\vec{F}_{1}.$$

## Homework Equations

My first attempt was to use the equivalence of the couples:
$$\vec{r}_{1}\times\vec{F}_{1}=\vec{r}_{2}\times\vec{F}_{2}=M_{1}\hat{k},$$
and the BAC - CAB rule for double cross product: $\vec A \times \vec B \times \vec C = \vec B (\vec A \ldotp \vec C) - \vec C (\vec A \ldotp \vec B)$

## The Attempt at a Solution

That $\vec{F}_1 = \vec{F}_2$, 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:
\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}So my point is: if there is freedom to choose both $\vec{r}_{2}$ and $\vec{F}_{2}$ , it's not possible to assume that these vectors should be related to the parameters of the problem, $\vec{r}_{1}$, $\vec{F}_{1}$ and $M_{1}\hat{k}$ as the author requires in his solution. Am I right or am I missing something really important on this issue?

Looks to me like you missed the easy part. If the net force on the system is originally F1, and the net force is to be held constant, then F2 = F1. Doesn't that enable you to finish your development of r2?

Well, basically, if $\vec{F}_1 = \vec{F}_2$, then it will be mandatory that $\vec{r}_1 = \vec{r}_2$, don't you think? Even if I put this reasoning forward, the result would be strange:
\begin{aligned}\vec{r}_{2} & =\frac{\left(\vec{F}_{1}\ldotp\vec{r}_{2}\right)\vec{F}_{1}+\vec{F}_{1}\times M_{1}\hat{k}}{F_{1}^{2}}\\ & =\frac{\left(\vec{F}_{1}\ldotp\vec{r}_{1}\right)\vec{F}_{1}+\vec{F}_{1}\times M_{1}\hat{k}}{F_{1}^{2}}\\ & =r_{1}\cos\theta\hat{F}_{1}+\frac{\vec{F}_{1}\times M_{1}\hat{k}}{F_{1}^{2}} \end{aligned}

Last edited:

## 1. What is a general equivalent couple - force system?

A general equivalent couple - force system is a combination of forces and couples that has the same effect on an object as an external force acting at a specific point. It is used to simplify complex systems and analyze the overall effect of the forces and couples on the object.

## 2. How is a general equivalent couple - force system different from a single force?

A single force is a force that acts on an object at a specific point, while a general equivalent couple - force system is a combination of forces and couples that can have the same effect as a single force. The difference lies in their magnitude, direction, and point of application.

## 3. How do you determine the general equivalent couple - force system of a complex system?

The general equivalent couple - force system can be determined by using the principle of transmissibility, which states that the external effect of a force is the same regardless of its point of application. This means that the forces and couples can be moved to a different point without changing the effect on the object.

## 4. What is the significance of using a general equivalent couple - force system in engineering?

In engineering, using a general equivalent couple - force system allows for the simplification of complex systems and makes it easier to analyze the overall effect of the forces and couples. It also helps in determining the stability and equilibrium of structures and machines.

## 5. Can a general equivalent couple - force system exist in three dimensions?

Yes, a general equivalent couple - force system can exist in three dimensions. In this case, the forces and couples act on different planes, and their effects are combined to determine the overall effect on the object. This is commonly used in analyzing the stability of three-dimensional structures.

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