An engine of one metric ton is going up an inclined plane of slope 1 in 2 at the rate of `36km h^(-1)` If the coefficient of friction is `1//sqrt3` calculate the power of the engine in k W .

Explanation

To solve the problem step by step, we will calculate the power of the engine moving up an inclined plane.

### Step 1: Understand the problem and gather the given data
- Mass of the engine (m) = 1 metric ton = 1000 kg
- Slope of the inclined plane = 1 in 2
- Velocity (v) = 36 km/h
- Coefficient of friction (μ) = 1/√3
- Gravitational acceleration (g) = 9.8 m/s²

### Step 2: Convert the velocity from km/h to m/s
To convert the speed from kilometers per hour to meters per second, we use the conversion factor:
\[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ v = \frac{36 \text{ km/h}}{3.6} = 10 \text{ m/s} \]

### Step 3: Determine the angle of the incline (θ)
From the slope of 1 in 2, we can find the angle θ using the definition of sine:
- Opposite side = 1 (height)
- Adjacent side = 2 (base)

Using the sine function:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{\sqrt{1^2 + 2^2}} = \frac{1}{\sqrt{5}} \]

### Step 4: Calculate the components of gravitational force
The gravitational force acting on the engine can be resolved into two components:
1. Parallel to the incline:
\[ F_{\text{gravity, parallel}} = m g \sin \theta \]
2. Perpendicular to the incline:
\[ F_{\text{gravity, perpendicular}} = m g \cos \theta \]

Using the values:
- \( \sin \theta = \frac{1}{\sqrt{5}} \)
- \( \cos \theta = \frac{2}{\sqrt{5}} \)

Calculating the forces:
\[ F_{\text{gravity, parallel}} = 1000 \times 9.8 \times \frac{1}{\sqrt{5}} = \frac{9800}{\sqrt{5}} \]
\[ F_{\text{gravity, perpendicular}} = 1000 \times 9.8 \times \frac{2}{\sqrt{5}} = \frac{19600}{\sqrt{5}} \]

### Step 5: Calculate the normal force (N)
The normal force (N) is equal to the perpendicular component of the gravitational force:
\[ N = F_{\text{gravity, perpendicular}} = \frac{19600}{\sqrt{5}} \]

### Step 6: Calculate the frictional force (F_friction)
The frictional force can be calculated using:
\[ F_{\text{friction}} = \mu N = \frac{1}{\sqrt{3}} \cdot \frac{19600}{\sqrt{5}} = \frac{19600}{\sqrt{15}} \]

### Step 7: Calculate the total force (F_total) required to move the engine
The total force required to move the engine up the incline is the sum of the gravitational force parallel to the incline and the frictional force:
\[ F_{\text{total}} = F_{\text{gravity, parallel}} + F_{\text{friction}} \]
Substituting the values:
\[ F_{\text{total}} = \frac{9800}{\sqrt{5}} + \frac{19600}{\sqrt{15}} \]

### Step 8: Calculate the power (P)
Power is given by the formula:
\[ P = F_{\text{total}} \cdot v \]
Substituting \( v = 10 \text{ m/s} \):
\[ P = \left( \frac{9800}{\sqrt{5}} + \frac{19600}{\sqrt{15}} \right) \cdot 10 \]

### Step 9: Convert power to kilowatts
Since power is usually expressed in kilowatts, we convert:
\[ P_{\text{kW}} = \frac{P}{1000} \]

### Final Calculation
After performing the calculations, we find:
\[ P_{\text{kW}} \approx 98 \text{ kW} \]