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A crate with a mass of [tex]$80.0 \, \text{kg}$[/tex] is on a ramp that is inclined at [tex]$32.0^{\circ}$[/tex].

What is the [tex][tex]$x$[/tex][/tex]-component of the weight of the crate?

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

Sagot :

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].