Question

Can someone help me with this

Answer 1

F = 360.6 N at -13.9° or 166.1°; The question may or may not expect either angle. It is the same angle, just expressed differently. My guess is that it wants the positive form of the angle.

Explanation:

We are given two force vectors and are asked to determine the magnitude of the resulting vector and the angle it makes with respect to the positive x-axis.

Lets denote our vectors:

Let's call the force on the right F₁ and the force on the left F₂.

• F₁ = 100 N at 60°

• F₂ = 400 N at 180°

** Note that the angle is measured counter-clockwise from the positive x-axis.

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To find the resultant force vector in magnitude-angle form from the two given force vectors, you can follow these steps:

Step 1: Convert the given force vectors in magnitude-angle form to vector form, using the following formulas:

For F₁:

• F₁_x = F₁cos(θ₁)

• F₁_y = F₁sin(θ₁)

Result ⇒ F₁ = <F₁_x, F₁_y>

For F₂:

• F₂_x = F₂cos(θ₂)

• F₂_y = F₂sin(θ₂)

Result ⇒ F₂ = <F₂_x, F₂_y>

Step 2: Add the corresponding components of the vectors to obtain the the resultant vector in vector form.

F = <F₁_x + F₂_x, F₁_y + F₂_y>

⇒ F = <F_x, F_y>

Step 3: Convert the resultant vector back to magnitude-angle form. To do this, use the following formulas:

• Magnitude: ||F|| = √((F_x)² + (F_y)²)

• Angle: θ = tan⁻¹(F_y/F_x) (Add 180° if F_x is negative)

`\hrulefill`

Step 1:

Given:

• F₁ = 100 N at 60°

• F₂ = 400 N at 180°

For F₁:

• F₁_x = (100)cos(60°) = 50 N

• F₁_y = (100)sin(60°) = 50√3 N

∴ F₁ = <50, 50√3> N

For F₂:

• F₂_x = (400)cos(180°) ≈ -400 N

• F₂_y = (400)sin(180°) ≈ 0 N

∴ F₂ = <-400, 0> N

Step 2:

Given:

• F₁ = <50, 50√3> N

• F₂ = <-400, 0> N

Adding x and y-components, we get:

⇒ F = <50 + (-400), 50√3 + 0>

∴ F = <-350, 50√3> N

Step 3:

Given:

• F_x = -350 N
• F_y = 50√3 N

Now finding the resultant vector in magnitude-angle form.

⇒ ||F|| = √((-350)² + (50√3)²)

⇒ ||F|| = 100√13

∴ ||F|| ≈ 360.6 N

⇒ θ = tan⁻¹(50√3 / -350)

∴ θ ≈ -13.9° or 166.1°

Therefore, the resulting force vector, F = 360.6 N at -13.9° or 166.1° (the question may or may not expect either angle).