Let's solve the problem step-by-step.
1. Identify the given values:
- The acceleration due to gravity, [tex]\( g = 9.8 \, \text{m/s}^2 \)[/tex]
- The angle, [tex]\( \theta = 30^\circ \)[/tex]
- The coefficient of friction, [tex]\( \mu = 0.30 \)[/tex]
2. Convert the angle from degrees to radians:
[tex]\[
\theta_{\text{rad}} = 30^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{6} \, \text{radians}
\][/tex]
3. Calculate the sine and cosine of the angle:
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
\cos(30^\circ) = \frac{\sqrt{3}}{2}
\][/tex]
4. Compute the components separately:
- Sin component:
[tex]\[
g \sin(30^\circ) = 9.8 \times \frac{1}{2} = 4.9 \, \text{m/s}^2
\][/tex]
- Cos component:
[tex]\[
\mu g \cos(30^\circ) = 0.30 \times 9.8 \times \frac{\sqrt{3}}{2} \approx 0.3 \times 9.8 \times 0.866 = 2.5461 \, \text{m/s}^2
\][/tex]
5. Subtract the cosine component from the sine component:
[tex]\[
4.9 - 2.5461 \approx 2.3539 \, \text{m/s}^2
\][/tex]
Thus, the final results are:
- The sine component: [tex]\( 4.9 \, \text{m/s}^2 \)[/tex]
- The cosine component: [tex]\( 2.5461 \, \text{m/s}^2 \)[/tex]
- The resulting value after subtraction: [tex]\( 2.3539 \, \text{m/s}^2 \)[/tex]
So, the steps clearly show how we arrive at the final values.
