A train of mass 1000000kg is drawn by an engine up an inclined plane 1 in 49 at 10mps , resistance due to friction is 1N for 1000kg find power of the engine .

To determine the power of the engine pulling the train up the inclined plane, we need to consider several factors, including the weight of the train, the incline of the plane, the speed of the train, and the frictional resistance. Let’s break this down step by step.

Understanding the Problem

We have a train with a mass of 1,000,000 kg, moving up an incline with a gradient of 1 in 49 at a speed of 10 m/s. The frictional resistance is given as 1 N for every 1,000 kg of mass. Our goal is to calculate the power required by the engine to overcome both the gravitational force and the frictional resistance.

Calculating the Forces Involved

• Weight of the Train: The weight (W) can be calculated using the formula W = m × g, where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²).
• Gravitational Force Along the Incline: The incline ratio of 1 in 49 means for every 49 units horizontally, the train rises 1 unit vertically. The angle θ can be found using the tangent function: tan(θ) = 1/49.
• Frictional Force: The total frictional force (F_friction) can be calculated as F_friction = (mass/1000) × 1 N.

Step-by-Step Calculations

Let’s perform the calculations one by one:

1. Calculate the Weight of the Train

W = 1,000,000 kg × 9.81 m/s² = 9,810,000 N

2. Determine the Gravitational Force Along the Incline

The force due to gravity acting down the slope (F_gravity) can be calculated as:

F_gravity = W × sin(θ)

To find sin(θ), we can use the relationship from the incline ratio:

sin(θ) = opposite/hypotenuse = 1/√(1² + 49²) = 1/√(2402) ≈ 0.0204

Thus, F_gravity = 9,810,000 N × 0.0204 ≈ 200,000 N

3. Calculate the Frictional Force

F_friction = (1,000,000 kg / 1000) × 1 N = 1,000 N

4. Total Force Required

The total force (F_total) that the engine must overcome is the sum of the gravitational force and the frictional force:

F_total = F_gravity + F_friction = 200,000 N + 1,000 N = 201,000 N

5. Calculate the Power of the Engine

Power (P) can be calculated using the formula:

P = F_total × v

Where v is the velocity of the train (10 m/s):

P = 201,000 N × 10 m/s = 2,010,000 W

Final Result

To convert watts to kilowatts (kW), we divide by 1,000:

P = 2,010,000 W / 1,000 = 2,010 kW

Therefore, the power of the engine required to pull the train up the inclined plane at the given speed, overcoming both gravitational and frictional forces, is approximately 2,010 kW.