If F_B = 560 N and F_C = 700 N, determine the magnitude and coordinate direction angles of the resultant force acting on the flag pole.
Image from: Hibbeler, R. C., S. C. Fan, Kai Beng. Yap, and Peter Schiavone. Statics: Mechanics for Engineers. Singapore: Pearson, 2013.
Solution:
Show me the final answers↓
We will first write the locations of points A, B and C are in Cartesian vector form.
Looking at the diagram, we can write the following:
B:(2i-3j+0k)
C:(3i+2j+0k)
We can now find the position vectors from A to B and A to C.
r_{AB}\,=\,\left\{2i-3j-6k\right\} m
r_{AC}\,=\,\left\{(3-0)i+(2-0)j+(0-6)k\right\}
r_{AC}\,=\,\left\{3i+2j-6k\right\} m
We now need to find the magnitude of each position vector so that we can use it to find the unit vector in the next step.
Magnitude of r_{AB}\,=\,7 m
Magnitude of r_{AC}\,=\,\sqrt{(3^2)+(2)^2+(-6)^2}
Magnitude of r_{AC}\,=\,7 m
Let us now find the unit vector for each force. Remember, this is found by dividing each component of the position vector by it’s magnitude.
u_{AC}\,=\,\left(\dfrac{3}{7}i\,+\,\dfrac{2}{7}j\,-\,\dfrac{6}{7}k\right)
We can now write force F_B and F_C in Cartesian form.
F_B\,=\,\left\{160i\,-\,240j\,-\,480k\right\} N
F_C\,=\,700\left(\dfrac{3}{7}i\,+\,\dfrac{2}{7}j\,-\,\dfrac{6}{7}k\right)
F_C\,=\,\left\{300i\,+\,200j\,-\,600k\right\} N
(Here we multiplied each individual component by the force. In other words, expand the brackets using FOIL)
We can find the resultant force by adding both forces together like so:
F_R\,=\,\left\{160i\,-\,240j\,-\,480k\right\}\,+\,\left\{300i\,+\,200j\,-\,600k\right\}
F_R\,=\,\left\{460i-40j-1080k\right\} N
To figure out the coordinate direction angles, we need to find the magnitude of the resultant force we just found.
magnitude of F_R\,=\,1174.56 N
Coordinate direction angles can be found taking the cosine inverse of each component of the resultant divided by the magnitude:
\beta\,=\,\cos^{-1}\left(\dfrac{-40}{1174.56}\right)\,=\,92^0
\gamma\,=\,\cos^{-1}\left(\dfrac{-1080}{1174.56}\right)\,=\,157^0
Final Answers:
\alpha\,=\,67^0
\beta\,=\,92^0
\gamma\,=\,157^0
Thank you for the answers
how about the height of the rope AB and Ac ? how to calculate it ?
I assume you mean lengths of each of the ropes? All you need to do is write a position vector from A to B, and then find the magnitude of it. That will give you the length. The same goes for rope AC, so you write a position vector from A to C, and then find the magnitude.