To find the [tex]$y$[/tex]-component of the weight of the box, we'll follow these steps:
1. Determine the weight of the box.
The weight [tex]$W$[/tex] of the box is given by the formula:
[tex]\[
W = m \times g
\][/tex]
where:
- [tex]\(m\)[/tex] is the mass of the box,
- [tex]\(g\)[/tex] is the acceleration due to gravity.
Given:
[tex]\[
m = 5.00 \text{ kg}
\][/tex]
[tex]\[
g = 9.81 \text{ m/s}^2
\][/tex]
Thus, the weight [tex]\(W\)[/tex] is:
[tex]\[
W = 5.00 \text{ kg} \times 9.81 \text{ m/s}^2 = 49.05 \text{ N}
\][/tex]
2. Resolve the weight into the [tex]$y$[/tex]-component.
The [tex]$y$[/tex]-component of the weight, denoted as [tex]\( w_y \)[/tex], is the component of the weight that acts perpendicular to the plane of the ramp.
Using trigonometry, the [tex]$y$[/tex]-component can be found using the cosine of the angle of inclination:
[tex]\[
w_y = W \times \cos(\theta)
\][/tex]
where:
- [tex]\( \theta \)[/tex] is the angle of the incline,
- [tex]\( W \)[/tex] is the weight of the box.
Given:
[tex]\[
\theta = 21.0^\circ
\][/tex]
Thus, the [tex]$y$[/tex]-component of the weight is:
[tex]\[
w_y = 49.05 \text{ N} \times \cos(21.0^\circ)
\][/tex]
3. Substitute the known values and calculate.
Using the cosine of [tex]\( 21.0^\circ \)[/tex]:
[tex]\[
\cos(21.0^\circ) \approx 0.93358
\][/tex]
Therefore:
[tex]\[
w_y = 49.05 \text{ N} \times 0.93358 \approx 45.79 \text{ N}
\][/tex]
Thus, the [tex]$y$[/tex]-component of the weight of the box is approximately:
[tex]\[
w_y \approx 45.79 \text{ N}
\][/tex]
So, the [tex]$y$[/tex]-component of the weight of the box is [tex]\( \boxed{45.79 \text{ N}} \)[/tex].
