One end of a wire of length L and weight w is attached rigidly to a point in roof and a weight `w_(1)` is suspended from its lower end. If A is the area of cross-section of the wire then the stress in the wire at a height `(3L)/(4)` from its lower end is

Options

A. `(w_(1))/(A)`
B. `(w_(1)+(w)/(4))/(A)`
C. `(w_(1)+(3w)/(4))/(A)`
D. `(w_(1)+w_(2))/(A)`

Answer

To solve the problem, we need to find the stress in the wire at a height of \( \frac{3L}{4} \) from its lower end. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Setup
We have a wire of length \( L \) attached to a roof, with a weight \( w_1 \) suspended from its lower end. The wire itself has a weight \( W \).

### Step 2: Identify the Point of Interest
We are interested in finding the stress at a point located at \( \frac{3L}{4} \) from the lower end of the wire.

### Step 3: Calculate the Weight of the Wire Segment
The total weight of the wire is \( W \). The weight per unit length of the wire is given by:
\[ \text{Weight per unit length} = \frac{W}{L} \]
To find the weight of the segment of the wire from the lower end to the point \( \frac{3L}{4} \), we calculate:
\[ \text{Weight of the segment} = \left(\frac{W}{L}\right) \times \left(\frac{3L}{4}\right) = \frac{3W}{4} \]

### Step 4: Determine the Total Force Acting at the Point
At the height \( \frac{3L}{4} \), the forces acting on the wire include:
1. The weight of the segment below this point, which is \( \frac{3W}{4} \).
2. The suspended weight \( w_1 \).

Thus, the total force \( F_{\text{net}} \) acting at this point is:
\[ F_{\text{net}} = \frac{3W}{4} + w_1 \]

### Step 5: Calculate the Stress
Stress is defined as the force per unit area. Therefore, the stress \( \sigma \) at the point \( \frac{3L}{4} \) from the lower end is given by:
\[ \sigma = \frac{F_{\text{net}}}{A} = \frac{\frac{3W}{4} + w_1}{A} \]

### Final Answer
Thus, the stress in the wire at a height \( \frac{3L}{4} \) from its lower end is:
\[ \sigma = \frac{3W}{4} + w_1 \div A \]