To find the x-component of a force directed at an angle, we use the concept of breaking down the vector into its components. The x-component of the force can be found using trigonometry, specifically the cosine function. Here's how to solve it step by step:
1. Understand the given quantities:
- The magnitude of the force, [tex]\( F \)[/tex], is [tex]\( 177 \, \text{N} \)[/tex].
- The angle, [tex]\( \theta \)[/tex], at which the force is applied relative to the horizontal is [tex]\( 85.0^\circ \)[/tex].
2. Convert the angle from degrees to radians:
Trigonometric functions in most contexts use angles in radians. To convert degrees to radians, we use the conversion factor:
[tex]\[
\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}
\][/tex]
Plugging in the given angle:
[tex]\[
\theta_{\text{radians}} = 85.0 \times \frac{\pi}{180} \approx 1.4835298641951802 \, \text{radians}
\][/tex]
3. Calculate the x-component of the force:
The x-component of the force, [tex]\( F_x \)[/tex], is calculated using the cosine of the angle. The formula is:
[tex]\[
F_x = F \cos(\theta_{\text{radians}})
\][/tex]
Substituting the known values:
[tex]\[
F_x = 177 \times \cos(1.4835298641951802)
\][/tex]
4. Evaluate the cosine function and the product:
Using the known cosine value for [tex]\( 1.4835298641951802 \)[/tex] radians:
[tex]\[
F_x = 177 \times 0.08729 \approx 15.42656646633549 \, \text{N}
\][/tex]
Therefore, the x-component of the force acting on the block is approximately [tex]\( 15.43 \, \text{N} \)[/tex].
