Two identical trains A and B move with equal actual speeds on parallel tracks along the equator. A moves 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 determine which train, A or B, exerts a greater force on the tracks, we can analyze the situation step by step.

### Step 1: Understand the Motion of the Trains
- Train A is moving from east to west, while Train B is moving from west to east. Both trains are moving at the same actual speed.

### Step 2: Consider the Earth's Rotation
- The Earth rotates from west to east. This means that at the equator, any object moving eastward (like Train B) is moving in the same direction as the Earth's rotation, while an object moving westward (like Train A) is moving against it.

### Step 3: Determine the Effective Velocity of Each Train
- The effective velocity of Train A (moving against the rotation) can be expressed as:
\[ V_A = V - \omega r \]
where \( V \) is the speed of the train, \( \omega \) is the angular velocity of the Earth, and \( r \) is the radius of the Earth.

- The effective velocity of Train B (moving with the rotation) can be expressed as:
\[ V_B = V + \omega r \]

### Step 4: Calculate the Forces Exerted by Each Train
- The normal force exerted by each train on the tracks can be affected by the centrifugal force due to the Earth's rotation.
- For Train A, the normal force \( N_A \) can be expressed as:
\[ N_A = mg - m \frac{V_A^2}{r} \]
Substituting for \( V_A \):
\[ N_A = mg - m \frac{(V - \omega r)^2}{r} \]

- For Train B, the normal force \( N_B \) can be expressed as:
\[ N_B = mg - m \frac{V_B^2}{r} \]
Substituting for \( V_B \):
\[ N_B = mg - m \frac{(V + \omega r)^2}{r} \]

### Step 5: Compare the Forces
- Since both trains have the same mass \( m \) and gravitational force \( mg \), we can compare the centrifugal forces:
- The centrifugal force for Train A is:
\[ F_{cA} = m \frac{(V - \omega r)^2}{r} \]
- The centrifugal force for Train B is:
\[ F_{cB} = m \frac{(V + \omega r)^2}{r} \]

- Since \( (V + \omega r)^2 \) is greater than \( (V - \omega r)^2 \), it follows that:
\[ F_{cB} > F_{cA} \]

### Conclusion
- Therefore, the normal force exerted by Train A on the tracks is greater than that exerted by Train B:
\[ N_A > N_B \]
- Hence, Train A exerts a greater force on the tracks than Train B.

### Final Answer
Train A will exert greater force on the tracks.

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