Final answer:
The x component of the force F₃ that balances F₁ and F₂ is -27.28 N, and the y component is 7.79 N. To find the angle of F₃ relative to the +x axis, we use the arctangent of the ratio of the y component to the x component, adjusting for the sign based on the quadrant in which F₃ lies.
Step-by-step explanation:
To find the force F₃ that balances the sum of forces F₁ and F₂, we first need to calculate the x and y components of the net force exerted by F₁ and F₂. For the x component of F₃, we have:
F₁_x + F₂_x + F₃_x = 0
8.88 N + 18.4 N + F₃_x = 0
F₃_x = - (8.88 N + 18.4 N)
F₃_x = -27.28 N
Similarly, for the y component:
F₁_y + F₂_y + F₃_y = 0
(-11.7 N) + 3.91 N + F₃_y = 0
F₃_y = -((-11.7 N) + 3.91 N)
F₃_y = 7.79 N
To find the angle relative to the +x axis, we use the arctangent function:
\(\theta = tan^{-1}\left(\frac{F₃_y}{F₃_x}\right)\)
\(\theta = tan^{-1}\left(\frac{7.79 N}{-27.28 N}\right)\)
\(\theta\) is calculated in the range of \(-180^\circ\) to \(180^\circ\). Since F₃ is in the second quadrant (negative x, positive y), we add \(180^\circ\) to the angle if it's negative, or subtract \(180^\circ\) if it's positive to find the correct directional angle.