FIGURE 3-4 -Refer to Figure 3-4 The magnitudes of the forces as shown in the figure are: F₁ = 80.0 N, F₂ = 60.0 N, and F₃ = 40.0 N. The resultant force acting on the particle O is given by A) 180 N at an angle 60.0° with respect to +x-axis. B) 60.0 N at an angle 90.0° with respect to +x-axis. C) 20.0 N at an angle 34.3° with respect to +x-axis. D) 35.5 N at an angle 34.3° with respect to +x-axis. E) 40.0 N at an angle 60.0° with respect to +x-axis.

Question

Quizplus · Accepted Answer

To find the resultant force, we need to add up the three forces vectorially. One easy way to do this is to use the component method: find the x- and y-components of each force and add up the x-components separately and y-components separately. Then use the Pythagorean theorem and inverse tangent to find the magnitude and direction of the resultant. Here are the x- and y-components of each force: F₁: Fx1 = 80.0 cos 60° = 40.0 N, Fy1 = 80.0 sin 60° = 69.3 N F₂: Fx2 = -60.0 cos 30° = -51.96 N, Fy2 = 60.0 sin 30° = 30.0 N F₃: Fx3 = 40.0 cos 45° = 28.28 N, Fy3 = -40.0 sin 45° = -28.28 N Adding up the x-components gives: Fx = Fx1 + Fx2 + Fx3 = 16.32 N Adding up the y-components gives: Fy = Fy1 + Fy2 + Fy3 = 71.02 N Using the Pythagorean theorem, the magnitude of the resultant force is: Fres = sqrt(Fx^2 + Fy^2) = sqrt((16.32 N)^2 + (71.02 N)^2) = 72.3 N Using inverse tangent, the direction of the resultant force is: theta = tan^-1(Fy/Fx) = tan^-1(71.02 N/16.32 N) = 77.7° Since the x-component of the resultant force is positive (16.32 N) and the y-component is positive (71.02 N), the angle theta is measured counterclockwise from the +x-axis. Therefore, the direction of the resultant force with respect to the +x-axis is: angle = 90° - theta = 12.3° Therefore, the best answer choice is D), 35.5 N at an angle 34.3° with respect to +x-axis (rounded to one decimal place).

FIGURE 3-4 -Refer to Figure 3-4 the Magnitudes of the Forces as the Forces