As shown in the figure,three force vectors act on an object.The magnitudes of the forces as shown in the figure are F₁ = 80.0 N,F₂ = 60.0 N,and F₃ = 40.0 N,where N is the standard SI unit of force.The resultant force acting on the object 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 the three force vectors by vector addition. One way to do this is by using the component method. First, we need to resolve each force vector into its x- and y-components.

F₁ = 80 N @ 120° = (-40 N, 69.3 N)
F₂ = 60 N @ 30° = (51.96 N, 30 N)
F₃ = 40 N @ 270° = (0 N, -40 N)

The @ symbol stands for "at an angle". For example, F₁ = 80 N @ 120° means that the magnitude of F₁ is 80 N and its direction is 120° counterclockwise from the +x-axis.

Now we can add the x- and y-components separately:
Fₓ = -40 N + 51.96 N + 0 N = 11.96 N
Fᵧ = 69.3 N + 30 N - 40 N = 59.3 N

The resultant force Fᵣ is the vector sum of Fₓ and Fᵧ:
|Fᵣ| = √(Fₓ² + Fᵧ²) = √(11.96² + 59.3²) = 60.9 N
θ = tan⁻¹(Fᵧ/Fₓ) = tan⁻¹(59.3/11.96) = 79.3°

However, this is not the final answer because the question asks for the angle with respect to the +x-axis, not with respect to the horizontal. To get the angle with respect to the +x-axis, we need to subtract 90° from the angle we just found:
θ' = 79.3° - 90° = -10.7°

This means that the resultant force is 60.9 N at an angle of 10.7° below the -x-axis. However, the question asks for the angle with respect to the +x-axis, so we need to add 180° (or subtract -180°) to this angle:
θ'' = -10.7° + 180° = 169.3°

So the final answer is 35.5 N at an angle of 169.3° with respect to the +x-axis (which is equivalent to saying at an angle of 10.7° with respect to the -x-axis). Therefore, the correct choice is D.

As Shown in the Figure,three Force Vectors Act on an Object.The