1) -7.14 N
2) +2.70 N
3) 7.63 N
Explanation:
1)
In order to find the x-component of the resultant force, we have to resolve each force along the x-axis.
The first force is 8.00 N and is directed at an angle of 61.0° above the negative x axis in the second quadrant: this means that the angle with respect to the positive x axis is
![180^{\circ}-61^{\circ}](https://tex.z-dn.net/?f=180%5E%7B%5Ccirc%7D-61%5E%7B%5Ccirc%7D)
so its x-component is
![F_{1x}=(8.00)(cos (180^{\circ}-61^{\circ}))=-3.88 N](https://tex.z-dn.net/?f=F_%7B1x%7D%3D%288.00%29%28cos%20%28180%5E%7B%5Ccirc%7D-61%5E%7B%5Ccirc%7D%29%29%3D-3.88%20N)
F₂ has a magnitude of 5.40 N and is directed at an angle of 52.8° below the negative x axis in the third quadrant: so, its angle with respect to the positive x-axis is
![180^{\circ}+52.8^{\circ}](https://tex.z-dn.net/?f=180%5E%7B%5Ccirc%7D%2B52.8%5E%7B%5Ccirc%7D)
Therefore its x-component is
![F_{2x}=(5.40)(cos (180^{\circ}+52.8^{\circ}))=-3.26 N](https://tex.z-dn.net/?f=F_%7B2x%7D%3D%285.40%29%28cos%20%28180%5E%7B%5Ccirc%7D%2B52.8%5E%7B%5Ccirc%7D%29%29%3D-3.26%20N)
So, the x-component of the resultant force is
![F_x=F_{1x}+F_{2x}=-3.88+(-3.26)=-7.14 N](https://tex.z-dn.net/?f=F_x%3DF_%7B1x%7D%2BF_%7B2x%7D%3D-3.88%2B%28-3.26%29%3D-7.14%20N)
2)
In order to find the y-component of the resultant force, we have to resolve each force along the y-axis.
The first force is 8.00 N and is directed at an angle of 61.0° above the negative x axis in the second quadrant: as we said previously, the angle with respect to the positive x axis is
![180^{\circ}-61^{\circ}](https://tex.z-dn.net/?f=180%5E%7B%5Ccirc%7D-61%5E%7B%5Ccirc%7D)
so its y-component is
![F_{1y}=(8.00)(sin (180^{\circ}-61^{\circ}))=7.00 N](https://tex.z-dn.net/?f=F_%7B1y%7D%3D%288.00%29%28sin%20%28180%5E%7B%5Ccirc%7D-61%5E%7B%5Ccirc%7D%29%29%3D7.00%20N)
F₂ has a magnitude of 5.40 N and is directed at an angle of 52.8° below the negative x axis in the third quadrant: as we said previously, its angle with respect to the positive x-axis is
![180^{\circ}+52.8^{\circ}](https://tex.z-dn.net/?f=180%5E%7B%5Ccirc%7D%2B52.8%5E%7B%5Ccirc%7D)
Therefore its y-component is
![F_{2y}=(5.40)(sin (180^{\circ}+52.8^{\circ}))=-4.30 N](https://tex.z-dn.net/?f=F_%7B2y%7D%3D%285.40%29%28sin%20%28180%5E%7B%5Ccirc%7D%2B52.8%5E%7B%5Ccirc%7D%29%29%3D-4.30%20N)
So, the y-component of the resultant force is
![F_y=F_{1y}+F_{2y}=7.00+(-4.30)=2.70 N](https://tex.z-dn.net/?f=F_y%3DF_%7B1y%7D%2BF_%7B2y%7D%3D7.00%2B%28-4.30%29%3D2.70%20N)
3)
The two components of the resultant force representent the sides of a right triangle, of which the resultant force corresponds tot he hypothenuse.
Therefore, we can find the magnitude of the resultant force by using Pythagorean's theorem:
![F=\sqrt{F_x^2+F_y^2}](https://tex.z-dn.net/?f=F%3D%5Csqrt%7BF_x%5E2%2BF_y%5E2%7D)
Where in this problem, we have:
is the x-component
is the y-component
And substituting, we find:
![F=\sqrt{(-7.14)^2+(2.70)^2}=7.63 N](https://tex.z-dn.net/?f=F%3D%5Csqrt%7B%28-7.14%29%5E2%2B%282.70%29%5E2%7D%3D7.63%20N)