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Which configuration of the slender bar AB, supported against a

smooth vertical wall by a massless string as shown in Fig.1(a), can

ensure static equilibrium. Explain your answer briefly with logic.



In the problem depicted, we have a slender bar \( AB \) supported by a smooth vertical wall at point \( A \), and a string of length \( L \) connected between points \( A \) and \( B \). The configurations differ in the angle formed by the string and the bar.

### Key Details:
- **Configuration 1**: The string forms an angle \( \theta \) with the bar.
- **Configuration 2**: The string is horizontal, making a \( 90^\circ \) angle with the vertical wall.

Now, to ensure **static equilibrium**, the sum of forces and moments acting on the system must be zero. Let's analyze both configurations:

### **Configuration 1 (Angle \( \theta \))**:

- **Forces**:
  1. Weight \( W \) acts vertically downwards at the center of mass of the bar.
  2. A normal reaction force \( N \) from the wall acts horizontally at point \( A \) (since the wall is smooth, there is no vertical component).
  3. Tension \( T \) in the string at point \( B \), having components both horizontally and vertically (due to angle \( \theta \)).

- **Moment Analysis**:
  To achieve static equilibrium, we take moments about point \( A \) (to eliminate the normal force \( N \)). The clockwise moment due to the weight \( W \) must be balanced by the counterclockwise moment from the tension \( T \). Given the angle \( \theta \), the string provides a vertical component of force to balance the weight. This configuration can achieve equilibrium if the tension and angle are appropriate to counteract the moment caused by the weight.

### **Configuration 2 (String at 90°)**:

- **Forces**:
  1. Weight \( W \) acts vertically downwards at the center of the bar.
  2. A normal reaction force \( N \) from the wall acts horizontally at point \( A \).
  3. Tension \( T \) in the string is purely horizontal, since the string is perpendicular to the wall.

- **Moment Analysis**:
  Taking moments about point \( A \), the tension in the string provides no vertical component to balance the weight of the bar, since it acts purely horizontally. Therefore, in this configuration, the system cannot be in static equilibrium because there is no upward force to counterbalance the downward weight \( W \). 

### **Conclusion**:
- **Configuration 1** can ensure static equilibrium because the tension in the string has both horizontal and vertical components, and the vertical component of tension can counteract the weight \( W \).
- **Configuration 2** cannot achieve equilibrium because the horizontal tension provides no vertical force to balance the bar's weight.

Would you like further details or clarifications? 

Here are 5 related questions to explore:
1. How do we calculate the exact tension required in Configuration 1 for equilibrium?
2. What are the conditions for equilibrium in terms of forces and moments for a general case like this?
3. How would friction at point \( A \) affect the equilibrium in either configuration?
4. What role does the center of mass of the bar play in determining the stability of these configurations?
5. How would changing the length of the string affect the equilibrium?

**Tip**: Always choose a point to sum the moments about which eliminates unknown forces like the normal force at point \( A \). This simplifies the analysis.

Which configuration of the slender bar AB, supported against a smooth vertical wall by a massless string as shown in Fig. 1(a), can ensure static equilibrium? Explain your answer briefly with logic.

Analyze the static equilibrium of a slender bar supported by a smooth vertical wall and a massless string. This problem covers key concepts such as force balance, moment calculations, and the conditions required for equilibrium. Suitable for undergraduate students studying mechanics or statics.

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