A cyclist moves along a curved road with a velocity v. The road is banked for speed v. The angle of banking is `theta`.Which of the following statements in not true?

Options

A. The cyclist will lean away from the vertical at an angle `theta`.
B. The normal reaction of the road will pass through the centre of gravity of the 'cycle plus cyclist' system
C. There will be no force of friction between the tyres and the road.
D. The cyclist is in equilibrium with respect to the ground.

Answer

To solve the question, we need to analyze each statement regarding the cyclist moving along a banked curve and determine which one is not true.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- A cyclist is moving along a curved road with a velocity \( v \).
- The road is banked at an angle \( \theta \) for speed \( v \).
- We need to evaluate four statements about the motion of the cyclist and identify which one is false.

2. **Evaluating Statement 1**:
- **Statement**: "The cyclist will lean away from the vertical at an angle \( \theta \)."
- **Analysis**: In a banked curve, the cyclist leans at an angle equal to the banking angle to maintain balance. This statement is **true**.

3. **Evaluating Statement 2**:
- **Statement**: "The normal reaction of the road will pass through the center of gravity of the cycle plus cyclist system."
- **Analysis**: The normal force acts vertically upwards through the center of gravity of the system, which is true for a cyclist on a banked road. This statement is **true**.

4. **Evaluating Statement 3**:
- **Statement**: "There will be no force of friction between the tires and the road."
- **Analysis**: On a properly banked curve, the banking angle provides the necessary centripetal force, and thus, ideally, there is no need for friction. This statement is **true**.

5. **Evaluating Statement 4**:
- **Statement**: "The cyclist is in equilibrium with respect to the ground."
- **Analysis**: The cyclist is moving in a curved path, which means the direction of velocity is changing. This implies that there is a net force acting on the cyclist (centripetal force), indicating that the cyclist is not in equilibrium. Therefore, this statement is **not true**.

6. **Conclusion**:
- The statement that is not true is **Option 4**: "The cyclist is in equilibrium with respect to the ground."

### Final Answer:
The statement that is not true is **Option 4**.

---