A tractor has its center of gravity located at a horizontal distance of $x_{cg} = 0.85\text{ m}$ ahead of the rear axle and at a vertical height of $y_{cg} = 0.75\text{ m}$ above the ground surface. Neglecting drawbar pull, the critical uphill slope angle ($\alpha_{\text{crit}}$) in degrees at which the tractor will experience static backward tipping (rearing of front wheels) is ________ (round off to 1 decimal place).
Answer Key:
48.6
Step-by-Step Solution & Explanation
```
Uphill Longitudinal Stability Limit:
Front wheels lift when the line of action of
weight W falls behind the rear tire contact patch.
tan(alpha_crit) = x_cg / y_cg
```
**Step-by-step Calculation:**
1. Moment equilibrium about rear tire contact point on slope $\alpha$:
$W \cos\alpha \times x_{cg} - W \sin\alpha \times y_{cg} = 0$
$\frac{\sin\alpha}{\cos\alpha} = \tan\alpha = \frac{x_{cg}}{y_{cg}}$
2. Substituting values:
$x_{cg} = 0.85\text{ m}, \quad y_{cg} = 0.75\text{ m}$
$\tan\alpha_{\text{crit}} = \frac{0.85}{0.75} = 1.1333$
3. Solving for critical slope angle:
$\alpha_{\text{crit}} = \tan^{-1}(1.1333) = 48.576^\circ \approx 48.6^\circ$