Q31. A train of mass 1000000 kg is drawn up an inclined plane 1in 49 at rate of 10mps by an engine, resistance due to friction is 1N for 1000kg of the train 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: the weight of the train, the incline of the plane, the frictional resistance, and the speed at which the train is moving. Let's break this down step by step.

Understanding the Forces at Play

The first step is to calculate the gravitational force acting on the train. The weight of the train can be calculated using the formula:

Weight (W) = mass (m) × gravitational acceleration (g)

Here, the mass of the train (m) is 1,000,000 kg, and the gravitational acceleration (g) is approximately 9.81 m/s². Thus:

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

Calculating the Component of Weight Along the Incline

The incline of the plane is given as 1 in 49, which means for every 49 units horizontally, the height increases by 1 unit. To find the angle of the incline, we can use the tangent function:

tan(θ) = opposite/adjacent = 1/49

Now, we can find the sine of the angle, which is necessary to calculate the component of the weight acting down the slope:

sin(θ) = 1/√(1² + 49²) = 1/√(1 + 2401) = 1/√2402 ≈ 0.0204

Now, the component of the weight acting down the incline (F_gravity) is:

F_gravity = W × sin(θ) = 9,810,000 N × 0.0204 ≈ 200,000 N

Frictional Resistance

The problem states that the resistance due to friction is 1 N for every 1000 kg of the train. Therefore, for a train of 1,000,000 kg:

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

Calculating the Total Force Required

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

F_total = F_gravity + F_friction = 200,000 N + 1000 N = 201,000 N

Finding the Power of the Engine

Power (P) is defined as the work done per unit time. The work done against the total force while moving at a constant speed can be calculated using the formula:

P = F_total × velocity

Given that the speed (velocity) of the train is 10 m/s, we can substitute the values:

P = 201,000 N × 10 m/s = 2,010,000 W

To convert this into kilowatts (kW), we divide by 1000:

P = 2,010,000 W / 1000 = 2010 kW

Final Result

The power of the engine required to pull the train up the inclined plane at a speed of 10 m/s, overcoming both gravitational and frictional forces, is approximately 2010 kW.