Answer:
Explanation:
We just need to sum the forces, we can do this easily in their Cartesian form.
Knowing the magnitude and angle with the positive x axis, we can find Cartesian representation of the vectors using the formula
[tex]\vec{A}= |\vec{A}| \ ( \ cos(\theta) \ , \ sin (\theta) \ )[/tex]
where [tex]|\vec{A}|[/tex] its the magnitude of the vector and θ the angle with the positive x axis.
So, for our forces we got:
[tex]\vec{F}_1 \ = \ 66 \ lbf \ * \ ( \ cos (30\°)\ , \ sin(30\°) \ )[/tex]
[tex]\vec{F}_2 \ = \ 110 \ lbf \ * \ ( \ cos (45\°)\ , \ sin(45\°) \ )[/tex]
[tex]\vec{F}_3 \ = \ 138 \ lbf \ * \ ( \ cos (120\°)\ , \ sin(120\°) \ )[/tex]
this will give us:
[tex]\vec{F}_1 \ = ( \ 57.157 \ lbf \ , \ 33 \ lbf \ )[/tex]
[tex]\vec{F}_2 \ = ( \ 77.782 \ lbf \ , \ 77.782 \ lbf \ )[/tex]
[tex]\vec{F}_3 \ = ( \ - 69 \ lbf \ , \ 119.511 \ lbf \ )[/tex]
Now, we just sum the forces:
[tex]\vec{F}_{net} \ = \ \vec{F}_1 \ + \ \vec{F}_2 \ + \ \vec{F}_3[/tex]
[tex]\vec{F}_{net} \ = ( \ 57.157 \ lbf \ , \ 33 \ lbf \ ) + ( \ 77.782 \ lbf \ , \ 77.782 \ lbf \ ) + (\ - 69 \ lbf \ , \ 119.511 \ lbf \ )[/tex]
[tex]\vec{F}_{net} \ = ( \ 57.157 \ lbf \ + \ 77.782 \ lbf \ - \ 69 \ lbf \, \ 33 \ lbf \ + \ 77.782 \ lbf \ + \ 119.511 \ lbf \ )[/tex]
[tex]\vec{F}_{net} \ = ( \ 65.939 \ lbf \, \ 230.293 lbf \ )[/tex]
This is the net force, to obtain the magnitude, we just need to find the length of the vector, using the Pythagorean formula:
[tex]|\vec{F}_{net}| = \sqrt{(F_{net_x})^2+(F_{net_y})^2}[/tex]
[tex]|\vec{F}_{net}| = \sqrt{(65.939 \ lbf)^2+(230.293 lbf)^2}[/tex]
[tex]|\vec{F}_{net}| = \ 239.547 \ lbf[/tex]
To obtain the angle with the positive x-axis we can use the formula:
[tex]\theta \ = \ arctan( \frac{F_y}{F_y})[/tex]
[tex]\theta \ = \ arctan( \frac{230.293 lbf}{65.939 \ lbf})[/tex]
[tex]\theta \ = \ arctan( 3.492)[/tex]
[tex]\theta \ = \ 74.02[/tex]
So, the answer its
[tex]magnitude = \ 239.547 \ lbf[/tex]
[tex]angle_{ (to the x axis)} = \ 74.02 [/tex]
Rounding up:
[tex]magnitude = \ 239.5 \ lbf[/tex]
[tex]angle_{ (to the x axis)} = \ 74.0 [/tex]
