To find the x-component of the weight of the box on a ramp that is inclined at [tex]\(15^{\circ}\)[/tex], we need to consider the component of the gravitational force parallel to the surface of the ramp. Follow these steps:
1. Determine the mass and angle:
- Mass of the box, [tex]\(m = 60.0 \, \text{kg}\)[/tex]
- Inclination angle, [tex]\(\theta = 15.0^{\circ}\)[/tex]
2. Understand the gravitational force:
- The gravitational force acts vertically downward and can be calculated using the formula:
[tex]\[
F_g = m \times g
\][/tex]
where [tex]\(g = 9.8 \, \text{m/s}^2\)[/tex] is the acceleration due to gravity.
3. Calculate the gravitational force:
[tex]\[
F_g = 60.0 \, \text{kg} \times 9.8 \, \text{m/s}^2 = 588 \, \text{N}
\][/tex]
4. Resolve the gravitational force into components:
- The x-component of the gravitational force (parallel to the ramp) can be found using:
[tex]\[
F_{gx} = F_g \times \sin(\theta)
\][/tex]
where [tex]\(\theta\)[/tex] must be in radians for precise calculations.
5. Convert the angle to radians:
- Since [tex]\(1 \, \text{degree} = \frac{\pi}{180} \, \text{radians}\)[/tex], convert the angle:
[tex]\[
\theta \, (\text{radians}) = 15.0 \times \frac{\pi}{180} \approx 0.2618 \, \text{radians}
\][/tex]
6. Calculate the x-component of the gravitational force:
- Now plug in the values:
[tex]\[
F_{gx} = 588 \, \text{N} \times \sin(0.2618) \approx 588 \times 0.2588 \approx 152.19 \, \text{N}
\][/tex]
So, the x-component of the weight of the box on the inclined ramp is [tex]\( \boxed{152.19 \, \text{N}} \)[/tex].
