Question

Two forces, F⃗ 1 and F⃗ 2, act at a point, as shown in the picture. (Figure 1) F⃗ 1 has a magnitude of 8.00 N and is directed at an angle of α = 61.0 ∘∘ above the negative x axis in the second quadrant. F⃗ 2 has a magnitude of 5.20 N and is directed at an angle of β = 54.1 ∘∘ below the negative x axis in the third quadrant.

What is the x component Fx of the resultant force? Express your answer in newtons.

What is the y component Fy of the resultant force? Express your answer in newtons.

What is the magnitude F of the resultant force? Express your answer in newtons.

What is the angle γγ that the resultant force forms with the negative x axis? In this problem, assume that positive angles are measured clockwise from the negative xx axis. Express your answer in degrees.

Answer 1

Answer 2

To find the components and magnitude of the resultant force, as well as the angle it forms with the negative x-axis, we can use vector addition. Let's break down the forces and solve step by step:

Given: F1 magnitude = 8.00 N F1 angle (α) = 61.0° (above the negative x-axis) F2 magnitude = 5.20 N F2 angle (β) = 54.1° (below the negative x-axis)

Step 1: Finding the x and y components of each force To find the x and y components, we use trigonometry. The x component (Fx) is given by F * cos(angle), and the y component (Fy) is given by F * sin(angle).

For F1: Fx1 = F1 * cos(α) = 8.00 N * cos(61.0°) ≈ -3.648 N (rounded to three decimal places)

Fy1 = F1 * sin(α) = 8.00 N * sin(61.0°) ≈ 6.956 N (rounded to three decimal places)

For F2: Fx2 = F2 * cos(β) = 5.20 N * cos(54.1°) ≈ -2.812 N (rounded to three decimal places)

Fy2 = F2 * sin(β) = 5.20 N * sin(54.1°) ≈ -4.207 N (rounded to three decimal places)

Step 2: Finding the resultant x and y components To find the resultant x and y components, we add the corresponding x and y components of the forces.

Fx = Fx1 + Fx2 = -3.648 N + (-2.812 N) ≈ -6.460 N (rounded to three decimal places)

Fy = Fy1 + Fy2 = 6.956 N + (-4.207 N) ≈ 2.749 N (rounded to three decimal places)

Step 3: Finding the magnitude and angle of the resultant force The magnitude (F) of the resultant force is given by the Pythagorean theorem: F = sqrt(Fx^2 + Fy^2).

F = sqrt((-6.460 N)^2 + (2.749 N)^2) ≈ 7.015 N (rounded to three decimal places)

The angle (γ) that the resultant force forms with the negative x-axis can be found using the inverse tangent function: γ = atan(Fy / Fx).

γ = atan(2.749 N / (-6.460 N)) ≈ -22.95° (rounded to two decimal places)

Note: The negative sign indicates that the angle is measured clockwise from the negative x-axis.

Summary: Fx ≈ -6.460 N Fy ≈ 2.749 N F ≈ 7.015 N γ ≈ -22.95°