One end of uniform wire of length `L` and of weight `W` is attached rigidly to a point in the roof and a weight `W_(1)` is suspended from its lower end. If `s` is the area of cross section of the wire, the stress in the wire at a height (`3L//4`) from its lower end is

Options

A. `(w_(1))/(S)`
B. `(w_(1)+(w)/(4))/(S)`
C. `((w_(1)+(3w)/(4))/(S))`
D. `(w_(1)+w)/(S)`

Explanation

To find the stress in the wire at a height of \( \frac{3L}{4} \) from its lower end, we can follow these steps:

### Step 1: Understand the Setup
We have a uniform wire of length \( L \) and weight \( W \) attached at one end to the roof. A weight \( W_1 \) is suspended from the lower end of the wire. We need to find the stress at a point that is \( \frac{3L}{4} \) from the lower end.

### Step 2: Identify the Forces Acting on the Wire
At the point \( \frac{3L}{4} \) from the lower end, the forces acting on the wire include:
1. The weight \( W_1 \) acting downward.
2. The weight of the portion of the wire below this point. Since the total weight of the wire is \( W \) and its length is \( L \), the weight of the portion of the wire from \( \frac{3L}{4} \) to \( L \) (which is \( \frac{L}{4} \)) can be calculated as:
\[ \text{Weight of the lower part of the wire} = \frac{1}{4} W \]

### Step 3: Calculate the Total Weight Acting Downward
The total downward force (restoring force) at the height \( \frac{3L}{4} \) is the sum of \( W_1 \) and the weight of the lower part of the wire:
\[ F = W_1 + \frac{1}{4} W \]

### Step 4: Calculate the Stress
Stress (\( \sigma \)) is defined as the force per unit area. The area of cross-section of the wire is given as \( S \). Therefore, the stress at the height \( \frac{3L}{4} \) from the lower end can be calculated as:
\[ \sigma = \frac{F}{S} = \frac{W_1 + \frac{1}{4} W}{S} \]

### Final Expression
Thus, the stress in the wire at a height of \( \frac{3L}{4} \) from the lower end is:
\[ \sigma = \frac{W_1 + \frac{1}{4} W}{S} \]