A flexible chain of weight W hangs between two fixed points A and B at the same level. The inclination of the chain with the horizontal at the two points of support is `theta`. Calculated the tension of the chain at end points and also at the lower most point .
[Hints : Consider equilibrium of each half of the chain. ]

Options

A. `(W)/(2).cosec theta`, `zero`
B. `(W)/(2).tan theta` , `zero`
C. `zero` ,`(W)/(2) cot theta`
D. none

Answer

To solve the problem of a flexible chain of weight \( W \) hanging between two fixed points \( A \) and \( B \) at the same level, with an inclination \( \theta \) at the points of support, we will analyze the forces acting on the chain.

### Step-by-Step Solution:

1. **Visualize the Setup**:
- Draw the chain hanging between points \( A \) and \( B \).
- The chain makes an angle \( \theta \) with the horizontal at both ends.

2. **Identify Forces**:
- The weight \( W \) of the chain acts downwards at the midpoint (the lowest point) of the chain.
- At points \( A \) and \( B \), the tension in the chain can be denoted as \( T \).

3. **Consider Equilibrium of Each Half of the Chain**:
- For the left half of the chain (from \( A \) to the midpoint \( C \)):
- The vertical component of the tension must balance the weight of the chain.
- The vertical component of tension at point \( A \) is \( T \sin \theta \).

4. **Balance Forces**:
- The total vertical force due to the tension from both sides (left and right) at the midpoint \( C \) is:
\[ 2T \sin \theta = W \]
- From this equation, we can solve for \( T \):
\[ T = \frac{W}{2 \sin \theta} \]

5. **Tension at the End Points**:
- The tension at points \( A \) and \( B \) is given by:
\[ T_A = T_B = \frac{W}{2 \sin \theta} \]

6. **Tension at the Lowermost Point \( C \)**:
- At the lowest point \( C \), the tension is affected by the forces from both sides.
- The horizontal components of tension from both sides cancel each other out, and since there are no vertical forces acting at point \( C \) (the weight \( W \) is balanced by the vertical components of tension), the net tension at point \( C \) is:
\[ T_C = 0 \]

### Final Results:
- Tension at points \( A \) and \( B \):
\[ T_A = T_B = \frac{W}{2 \sin \theta} \]
- Tension at the lowermost point \( C \):
\[ T_C = 0 \]