Question

A train consists of an engine and 3 coaches, first coach is closest to the engine, third coach is farthest from engine. The train is moving with a constant acceleration a. The mass of each coach is M. The force exerted by the first coach on the second coach will be:

• A. Ma
• B. 2Ma
• C. 3Ma
• D. 4Ma
• E. \(\sqrt{2}\)Ma

Answer 1

The Correct Option is B

Given:

• Train configuration: 1 engine + 3 coaches (Coach 1 closest to engine, Coach 3 farthest)
• Constant acceleration of train: \( a \)
• Mass of each coach: \( M \)

Step 1: Analyze Forces on Coach 2

For Coach 2 to accelerate at \( a \), the net force acting on it must be \( Ma \). This force is provided by:

1. The pull from Coach 1 (let's call this \( F_{1 \rightarrow 2} \))
2. The backward force from Coach 3 (which resists motion)

Step 2: Consider Coach 3's Resistance

Coach 3 has mass \( M \) and must also accelerate at \( a \), requiring force \( Ma \). This force comes entirely from Coach 2 pushing it forward (\( F_{2 \rightarrow 3} = Ma \)).

By Newton's 3rd Law, Coach 3 exerts an equal and opposite force on Coach 2: \( F_{3 \rightarrow 2} = -Ma \).

Step 3: Apply Newton's 2nd Law to Coach 2

The net force on Coach 2 is:

\[ F_{1 \rightarrow 2} + F_{3 \rightarrow 2} = Ma \]

Substitute \( F_{3 \rightarrow 2} \):

\[ F_{1 \rightarrow 2} - Ma = Ma \]

\[ F_{1 \rightarrow 2} = 2Ma \]

Conclusion:

The force exerted by the first coach on the second coach is \( 2Ma \).

Answer: \(\boxed{B}\)

Answer 2

To determine the force exerted by the first coach on the second coach in a train moving with a constant acceleration \( a \), we can analyze the forces acting on each coach individually. Here is the step-by-step solution:

1. Understanding the setup:
  - The train consists of an engine and 3 coaches.
  - Each coach has mass \( M \).
  - The train is accelerating with acceleration \( a \).

2. Analyzing forces on the third coach:
  - The third coach is the farthest from the engine and only experiences the force exerted by the second coach.
  - Let's denote the force exerted by the second coach on the third coach as \( F_{23} \).
  - Using Newton's second law for the third coach:
    \[ F_{23} = Ma \]

3. Analyzing forces on the second coach:
  - The second coach experiences the force exerted by the first coach (denoted as \( F_{12} \)) and the reaction force from the third coach \( -F_{23} \).
  - Using Newton's second law for the second coach:
    \[ F_{12} - F_{23} = Ma \]
  - Substituting \( F_{23} = Ma \) into the equation:
    \[ F_{12} - Ma = Ma \]
    \[ F_{12} = Ma + Ma \]
    \[ F_{12} = 2Ma \]

Thus, the force exerted by the first coach on the second coach is \( 2Ma \).
So, The correct answer is Option (B): 2Ma

Answer 3

1. Visualize the system:

We have an engine pulling three coaches. The first coach is closest to the engine, and the third coach is farthest. The train is accelerating with a constant acceleration 'a'.

2. Consider the forces on the coaches:

The engine pulls all three coaches. The first coach pulls the second and third coaches. The second coach pulls the third coach.

3. Focus on the force exerted by the first coach on the second coach:

The first coach needs to accelerate its own mass and the mass of the two coaches behind it. The total mass being pulled by the first coach is 2M (the second and third coaches).

4. Apply Newton's second law:

Force (F) = mass (m) × acceleration (a)

In this case, the mass being accelerated by the force exerted by the first coach is 2M, and the acceleration is 'a'. Therefore:

\[F = (2M) \times a = 2Ma\]