The pipe assembly is subjected to


The pipe assembly is subjected to the 80-N force. Determine the moment of this force about point A.

The pipe assembly is subjected to

Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.

Solution:

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When trying to find the moment in 3-D coordinates,  the first step is to express the force in Cartesian vector form. Let us express the 80 N in Cartesian form.

The pipe assembly is subjected to

To express this force in Cartesian form, we first need to find F’. F’ is the force that lies on the x-y plane. We then use that to calculate F_y and F_x. Note that this is a step that must always be performed if a force is shown in that manner. From the diagram, we see that:

F'=80\cos30^0=69.28 N

 

Notice how F’ is now the hypotenuse of the new triangle that lies on the x-y plane. Again, using F’ as the hypotenuse, we can easily calculate F_y and F_x.

F_x=69.28\sin40^0=44.5 N

F_y=69.28\cos40^0=53.1 N

 

F_z can be found by taking the \sin30^0 of 80 N.

F_z=-80\sin30^0=-40 N
(Why is F_z negative? Notice how F_z is downwards. In other words, it’s along the negative z-axis.)

 

We can now express the force in Cartesian form:

F=\left\{44.5i+53.1j-40k\right\}

 

The next step is to express a position vector from where we want to calculate the moment to where the force is being applied. In this question, we are trying to find the moment at A, thus our position vector begins at A. The force is applied at C. Therefore, our position vector would be from A to C.

The pipe assembly is subjected position vector

To express the position vector, we need to figure out where points A and C lie in our diagram. We write them both down in Cartesian form. From the diagram, we see that:

A:(0i+0j+0k)\text{m}

C:((300+250)i+400j-200k)\text{mm}=(0.55i+0.4j-0.2k)\text{m}

 

Thus, r_{AC} is:

r_{AC}=\left\{(0.55-0)i+(0.4-0)j+(-0.2-0)k\right\}

r_{AC}=\left\{0.55i+0.4j-0.2k\right\}

A position vector, denoted \mathbf{r} is a vector beginning from one point and extending to another point. It is calculated by subtracting the corresponding vector coordinates of one point from the other. If the coordinates of point A was (x_A,y_A,z_A) and the coordinates of point B was(x_B,y_B,z_B), then r_{AB}\,=\,(x_B-x_A)i+(y_B-y_A)j+(z_B-z_A)k

 

The final step is to take the cross product of the position vector and the force, which will give us the moment produced at A.

M_A=r_{AC}\times F

M_A=\begin{bmatrix}\bold i&\bold j&\bold k\\0.55&0.4&-0.2\\44.5&53.1&-40\end{bmatrix}

M_A=\left\{-5.38i+13.1j+11.4k\right\}\,\text{N}\cdot\text{m}

 

Final Answer:

M_A=\left\{-5.38i+13.1j+11.4k\right\}\,\text{N}\cdot\text{m}

 

This question can be found in Engineering Mechanics: Statics (SI edition), 13th edition, chapter 4, question 4-40.

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