A rear-wheel-drive tractor has a total static mass of $M = 3000\text{ kg}$ and a wheelbase of $L = 2.20\text{ m}$. When the tractor rests on level platform scales, the static reaction on the front wheels is measured to be $W_f = 10.5\text{ kN}$. Taking $g = 9.81\text{ m/s}^2$, the horizontal longitudinal distance from the rear axle centerline to the tractor center of gravity ($x_{cg}$) in $\text{m}$ is ________ (round off to 2 decimal places).
Answer Key:
0.78
Step-by-Step Solution & Explanation
```
Static Tractor Chassis Equilibrium:
Front Axle Rear Axle
Reaction W_f Reaction W_r
| |
+============================+
|<-- x_cg -->| (CG)
|<------ Wheelbase L = 2.20 m ------->|
```
**Step-by-step Calculation:**
1. Total static weight of tractor ($W$):
$W = M \times g = 3000\text{ kg} \times 9.81\text{ m/s}^2 = 29430\text{ N} = 29.43\text{ kN}$
2. Taking moments about the rear axle ground contact point:
$\sum M_{\text{rear}} = 0 \implies (W \times x_{cg}) - (W_f \times L) = 0$
$x_{cg} = \frac{W_f \times L}{W}$
3. Substituting values:
$W_f = 10.5\text{ kN}, \quad L = 2.20\text{ m}, \quad W = 29.43\text{ kN}$
$x_{cg} = \frac{10.5 \times 2.20}{29.43} = \frac{23.10}{29.43} = 0.7849\text{ m} \approx 0.78\text{ m}$