The lamp has a mass of 15 kg and is supported by a pole AO and cables AB and AC. If the force in the pole acts along its axis, determine the forces in AO, AB, and AC for equilibrium.

#### Solution:

Show me the final answer↓

We will express each force in the pole and cables in Cartesian vector form. To do so, we will find the locations of the points in Cartesian vector form.

Using the diagram, the locations of the points A, B, C, and O are:

B:(-4i+1.5j+0k)

C:(0i+1.5j+0k)

O:(0i+0j+0k)

The position vectors for points from A to B, A to C, and A to O are:

r_{AC}\,=\,\left\{(0-2)i+(1.5-(-1.5))j+(0-6)k\right\}\,=\,\left\{-2i+3j-6k\right\}

r_{OA}\,=\,\left\{(2-0)i+(-1.5-0)j+(6-0)k\right\}\,=\,\left\{2i-1.5j+6k\right\}

*(Why did we write the position vector from O to A instead of A to O? Unlike ropes, which can only be in tension, the pole will actually be in compression. That means a force from the pole is heading upwards, where as in the cables, the force is heading away from the lamp)*

The magnitude of each position vector is:

magnitude of r_{AC}\,=\,\sqrt{(-2)^2+(3)^2+(-6)^2}\,=\,7

magnitude of r_{OA}\,=\,\sqrt{(2)^2+(-1.5)^2+(6)^2}\,=\,6.5

The unit vector for each position vector is:

u_{AC}\,=\,\left(-\dfrac{2}{7}i+\dfrac{3}{7}j-\dfrac{6}{7}k\right)

u_{OA}\,=\,\left(\dfrac{2}{6.5}i-\dfrac{1.5}{6.5}j+\dfrac{6}{6.5}k\right)

We can now write each force in Cartesian vector form:

F_{AC}\,=\,F_{AC}\left(-\dfrac{2}{7}i+\dfrac{3}{7}j-\dfrac{6}{7}k\right)

F_{OA}\,=\,F_{OA}\left(\dfrac{2}{6.5}i-\dfrac{1.5}{6.5}j+\dfrac{6}{6.5}k\right)

(Further simplify by expanding the brackets and writing the fractions in decimal form)

F_{AB}\,=\,\left\{-0.667F_{AB}i+0.333F_{AB}j-0.667F_{AB}k\right\}

F_{AC}\,=\,\left\{-0.286F_{AC}i+0.429F_{AC}j-0.857F_{AC}k\right\}

F_{OA}\,=\,\left\{0.308F_{OA}i-0.231F_{OA}j+0.923F_{OA}k\right\}

F\,=\,\left\{0i+0j-(15)(9.81)\right\}\,=\,\left\{0i+0j-147.15k\right\}

(Force F is the weight of the lamp, which only has a z-component)

Since the system is in equilibrium, all forces added together must equal zero.

F_{AB}+F_{AC}+F_{OA}+F\,=\,0

Furthermore, as each force added together must equal zero, than each individual component (x, y, z-components) added together must also equal zero.

-0.667F_{AB}-0.286F_{AC}+0.308F_{OA}\,=\,0

y-components:

0.333F_{AB}+0.429F_{AC}-0.231F_{OA}\,=\,0

z-components:

-0.667F_{AB}-0.857F_{AC}+0.923F_{OA}-147.15\,=\,0

Solving the three equations simultaneously gives us:

F_{AC}\,=\,86.1 N

F_{OA}\,=\,319.2 N

Thank you for the awesome work you all do.

I have 1 question?

How did you come up with forces that are in the green box?

Thanks

Hi,

The green box answers are found by solving the component equations we found (in the blue box above). There are three equations with three unknowns and we can solve for those three. When you solve them, you will end up with the values for each force in the cables. All steps are shown to the point where you can get the three equations to solve. If you don’t remember how to solve them, please kindly look at this link: https://goo.gl/IFzQTd

If you are still unclear, let us know, and we will try to clear it up some more!

Thanks!

Eureka! Found the solutions. Thank you so very much for the assistance and guidance. This is a very lengthy problem to solve with an even longer system of equations solution.

Glad to see you got it! Yes, it is a long and tedious task, but usually, with these types of questions, (where we don’t calculate the moment), you can follow a series of steps to arrive at the answer. They are usually as follows:

1) Write down the locations of all the points

2) Find the position vector for each cable/force with respect to the object

3) Find the magnitude of each position vector

4) Use the magnitude to find the unit vector

5) Write each force in Cartesian vector form using the unit vector you found

6) Solve the system of equations to find the forces

Anyways, you are very welcome, best of luck in your studies!

Any know how to get the final value

I watched the video that’s mentioned in the comments still didn’t understand

Please let us know which part you have a hard time understanding? We will do our best to explain it to you. 🙂

I don’t understand where you are getting the 9.81 to multiply with the weight of the lamp from?

Hi. 9.81 is the acceleration due to gravity. Remember that in the question, it tells us that the

massof the lamp is 15 kg, not theweight. To find the weight, we need to multiply the mass by the acceleration due to gravity. For more information, please read: http://bit.ly/2EqvBILThank you very much 🙂

A great solution. Helping me understand forces better. But I have another question in retion to this. Determine the moment about B induced by the weight of the lamp acting at A. A little help would be appreciated

Please see the following video where I explain everything about moments, including how to solve 3D problems: https://www.youtube.com/watch?v=QNNnPZ68STI

Many thanks!