Sure! Let's solve this step-by-step.
1. Understand the problem:
We need to find the x-component of the weight of a crate that is on an incline. The incline angle is given as [tex]\( 32.0^{\circ} \)[/tex], and the mass of the crate is [tex]\( 80.0 \, \text{kg} \)[/tex].
2. Identify given values:
- Mass of the crate ([tex]\( m \)[/tex]) = [tex]\( 80.0 \, \text{kg} \)[/tex]
- Incline angle ([tex]\( \theta \)[/tex]) = [tex]\( 32.0^{\circ} \)[/tex]
- Acceleration due to gravity ([tex]\( g \)[/tex]) = [tex]\( 9.81 \, \text{m/s}^2 \)[/tex]
3. Convert the angle from degrees to radians:
To perform trigonometric calculations, we generally use radians. We convert degrees to radians using the formula:
[tex]\[
\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}
\][/tex]
Substituting [tex]\( \theta_{\text{deg}} = 32.0 \)[/tex]:
[tex]\[
\theta_{\text{rad}} = 32.0 \times \frac{\pi}{180} \approx 0.5585 \, \text{radians}
\][/tex]
4. Calculate the weight of the crate:
The weight ([tex]\( w \)[/tex]) is given by the mass times the gravitational acceleration:
[tex]\[
w = m \times g
\][/tex]
Substituting [tex]\( m = 80.0 \, \text{kg} \)[/tex] and [tex]\( g = 9.81 \, \text{m/s}^2 \)[/tex]:
[tex]\[
w = 80.0 \times 9.81 = 784.8 \, \text{N}
\][/tex]
5. Calculate the x-component of the weight:
The x-component of the weight ([tex]\( w_x \)[/tex]) can be found using the sine of the incline angle:
[tex]\[
w_x = w \times \sin(\theta_{\text{rad}})
\][/tex]
Substituting [tex]\( w = 784.8 \, \text{N} \)[/tex] and [tex]\( \sin(0.5585) \approx 0.5299 \)[/tex]:
[tex]\[
w_x = 784.8 \times 0.5299 \approx 415.88 \, \text{N}
\][/tex]
Therefore, the x-component of the weight of the crate is approximately [tex]\( 415.88 \, \text{N} \)[/tex].
