High School

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A chair is pulled by two horizontal forces. The first force is 122 N at an angle of [tex]$43.6^{\circ}$[/tex], and the second is 97.6 N at an angle of [tex]$49.9^{\circ}$[/tex].

What is the x-component of the total force acting on the chair?

[tex]\[ \overrightarrow{F_x} = [?] \, \text{N} \][/tex]

Answer :

Sure, let's break down the process of finding the x-component of the total force acting on the chair step-by-step.

### Step-by-step Solution:

1. Understanding the Problem:
The chair is subjected to two horizontal forces at different angles. We need to determine the x-component (horizontal part) of these forces and then sum them to find the total x-component of the force acting on the chair.

2. Given Information:
- Force 1 ([tex]\( F_1 \)[/tex]) = 122 N
- Angle for Force 1 ([tex]\( \theta_1 \)[/tex]) = 43.6 degrees
- Force 2 ([tex]\( F_2 \)[/tex]) = 97.6 N
- Angle for Force 2 ([tex]\( \theta_2 \)[/tex]) = 49.9 degrees

3. Decomposing Forces into Components:
We need to calculate the x-components of each force using the formula:
[tex]\[
F_{x} = F \cdot \cos(\theta)
\][/tex]
Here, [tex]\( F \)[/tex] is the magnitude of the force and [tex]\( \theta \)[/tex] is the angle of the force.

4. Calculating the x-component for Force 1:
[tex]\[
F_{1x} = 122 \cdot \cos(43.6^\circ)
\][/tex]

5. Calculating the x-component for Force 2:
[tex]\[
F_{2x} = 97.6 \cdot \cos(49.9^\circ)
\][/tex]

6. Summing the x-components:
The total x-component is the sum of the individual x-components:
[tex]\[
\overrightarrow{F_x} = F_{1x} + F_{2x}
\][/tex]

7. Numerical Results:
The calculated values are:
- [tex]\( F_{1x} \approx 88.35 \)[/tex] N
- [tex]\( F_{2x} \approx 62.87 \)[/tex] N

Therefore, the total x-component of the force acting on the chair is:
[tex]\[
\overrightarrow{F_x} = 88.35 \text{ N} + 62.87 \text{ N} = 151.22 \text{ N}
\][/tex]

### Final Answer:
The x-component of the total force acting on the chair is approximately [tex]\( 151.22 \)[/tex] N.

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