A truck of amss 10 ton is movng up an incline of 1 in 50 with a uniform velocity of 72 km/h. If the resistance opposing the motion of the truck due to friction is 5 kg per ton, the power of the engine is `(g=10m//s^(2))`

Options

A. 35kW
B. 40kW
C. 45kW
D. 50kW

Explanation

To solve the problem, we need to calculate the power of the engine required to move the truck up the incline with a uniform velocity. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Given Data
- Mass of the truck (m) = 10 tons = 10 × 1000 kg = 10000 kg
- Incline ratio = 1 in 50 (this means for every 50 units of horizontal distance, the height increases by 1 unit)
- Velocity (v) = 72 km/h = \( \frac{72 \times 1000}{3600} \) m/s = 20 m/s
- Resistance due to friction = 5 kg per ton = 5 × 10 tons = 50 kg
- Gravitational acceleration (g) = 10 m/s²

### Step 2: Calculate the Forces Acting on the Truck
1. **Weight of the Truck (W)**:
\[ W = m \cdot g = 10000 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 100000 \, \text{N} \]

2. **Component of Weight along the Incline (F_gravity)**:
\[ F_{\text{gravity}} = W \cdot \sin(\theta) \]
Since the incline is 1 in 50, we can find \( \sin(\theta) \) as:
\[ \sin(\theta) = \frac{1}{\sqrt{1^2 + 50^2}} \approx \frac{1}{50} \]
Therefore,
\[ F_{\text{gravity}} = 100000 \cdot \frac{1}{50} = 2000 \, \text{N} \]

3. **Frictional Force (F_friction)**:
\[ F_{\text{friction}} = \text{mass of truck} \cdot \text{friction per ton} \cdot g = 50 \, \text{kg} \cdot 10 \, \text{m/s}^2 = 500 \, \text{N} \]

### Step 3: Calculate the Total Force Required (F_total)
The total force required to move the truck up the incline at constant velocity is the sum of the gravitational component and the frictional force:
\[ F_{\text{total}} = F_{\text{gravity}} + F_{\text{friction}} = 2000 \, \text{N} + 500 \, \text{N} = 2500 \, \text{N} \]

### Step 4: Calculate the Power of the Engine
Power (P) is given by the formula:
\[ P = F_{\text{total}} \cdot v \]
Substituting the values we have:
\[ P = 2500 \, \text{N} \cdot 20 \, \text{m/s} = 50000 \, \text{W} = 50 \, \text{kW} \]

### Final Answer
The power of the engine is **50 kW**.

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