To solve for the [tex]\( y \)[/tex]-component of the weight of a 150 kg crate on a ramp inclined at [tex]\( 18.0^\circ \)[/tex], let's follow these steps:
1. Calculate the weight of the crate:
- The weight ([tex]\( w \)[/tex]) is determined by the mass ([tex]\( m \)[/tex]) and gravitational acceleration ([tex]\( g \)[/tex]).
- Given: [tex]\( m = 150 \)[/tex] kg and [tex]\( g = 9.81 \)[/tex] m/s[tex]\(^2\)[/tex].
Formula for weight:
[tex]\[
w = m \cdot g
\][/tex]
Substituting the given values:
[tex]\[
w = 150 \, \text{kg} \times 9.81 \, \text{m/s}^2 = 1471.5 \, \text{N}
\][/tex]
2. Convert the angle to radians:
- The angle given is [tex]\( 18.0^\circ \)[/tex].
- We convert degrees to radians using the formula:
[tex]\[
\text{radians} = \text{degrees} \times \left( \frac{\pi}{180} \right)
\][/tex]
Substituting the given angle:
[tex]\[
\text{radians} = 18.0 \times \left( \frac{\pi}{180} \right) = 0.3141592653589793 \, \text{radians}
\][/tex]
3. Calculate the [tex]\( y \)[/tex]-component of the weight:
- The [tex]\( y \)[/tex]-component of the weight ([tex]\( w_y \)[/tex]) is found by multiplying the weight by the sine of the angle in radians.
Formula for the [tex]\( y \)[/tex]-component of the weight:
[tex]\[
w_y = w \cdot \sin(\text{angle in radians})
\][/tex]
Substituting the given values:
[tex]\[
w_y = 1471.5 \, \text{N} \times \sin(0.3141592653589793)
\][/tex]
- Using a calculator to find the sine value (assuming a pre-calculated value):
[tex]\[
w_y = 1471.5 \, \text{N} \times 0.30901699437494745 = 454.71850722273507 \, \text{N}
\][/tex]
So, the [tex]\( y \)[/tex]-component of the weight of the crate is approximately:
[tex]\[
w_y = 454.72 \, \text{N}
\][/tex]
