Two identical trains A and B move with equal actual speeds on parallel tracks along the equator. A moves from east to west and B, from west to east. Which train will exert greater force on the tracks?

Options

A. A
B. B
C. They will exert equal force
D. The mass and the speed of each train must be known to reach a conclusion

Answer

To solve the problem of which train exerts a greater force on the tracks, we need to analyze the forces acting on each train due to their motion relative to the Earth's rotation. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have two identical trains, A and B, moving on parallel tracks along the equator. Train A moves from east to west, while train B moves from west to east. Both trains have the same actual speed.

### Step 2: Define Variables
- Let \( V \) be the actual speed of both trains.
- Let \( \omega \) be the angular velocity of the Earth.
- Let \( r \) be the radius of the Earth.

### Step 3: Calculate the Effective Velocity
For Train A (moving from east to west):
- The effective velocity \( V_A \) relative to the ground is given by:
\[ V_A = V - \omega r \]

For Train B (moving from west to east):
- The effective velocity \( V_B \) relative to the ground is given by:
\[ V_B = V + \omega r \]

### Step 4: Determine the Forces
The force exerted by each train on the tracks can be analyzed using the concept of apparent weight, which is influenced by the centripetal force required for circular motion due to the Earth's rotation.

The normal force \( N \) exerted by the tracks on the trains can be calculated using the following equations:

For Train A:
\[ N_A = mg - m \cdot \frac{(V - \omega r)^2}{r} \]

For Train B:
\[ N_B = mg - m \cdot \frac{(V + \omega r)^2}{r} \]

### Step 5: Compare the Forces
To determine which train exerts a greater force, we need to compare \( N_A \) and \( N_B \).

1. **Expand the equations**:
- For Train A:
\[ N_A = mg - m \cdot \frac{(V^2 - 2V\omega r + \omega^2 r^2)}{r} \]
- For Train B:
\[ N_B = mg - m \cdot \frac{(V^2 + 2V\omega r + \omega^2 r^2)}{r} \]

2. **Simplify and compare**:
- The term \( mg \) is common in both equations, so we focus on the terms involving \( V \) and \( \omega \).
- It can be observed that \( N_A \) will be greater than \( N_B \) because the negative contribution from the centripetal term is larger for Train B due to the addition of \( 2V\omega r \).

### Conclusion
Since \( N_A > N_B \), Train A (moving from east to west) exerts a greater force on the tracks compared to Train B.

### Final Answer
**Train A exerts a greater force on the tracks.**

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