Question

A uniform wire has length \(L\) and weight \(W\). One end of the wire is attached rigidly to the roof and weight \(W_1\) is suspended from its lower end. If \(A\) is the cross-sectional area of the wire, then the stress in the wire at a height \(\dfrac{3L}{4}\) from its lower end is

• A. \(\dfrac{4W_1 + 3W}{4A}\)
• B. \(\dfrac{3W_1 - 4W}{2A}\)
• C. \(\dfrac{3W_1 + 4W}{2A}\)
• D. \(\dfrac{4W_1 - 3W}{4A}\)

Hint:

Stress at any point in a hanging wire equals total weight suspended below that point divided by cross-sectional area.

Answer 1

The Correct Option is A

Step 1: Understand the forces acting.
Stress at a point in the wire depends on the load supported below that point.

Step 2: Weight of wire below the given point.
Height from lower end = \(\dfrac{3L}{4}\).
So length of wire below this point = \(\dfrac{3L}{4}\).
Weight of this part of wire = \(\dfrac{3W}{4}\).

Step 3: Total force acting at that section.
\[ F = W_1 + \frac{3W}{4} \]
Step 4: Stress calculation.
\[ \text{Stress} = \frac{F}{A} = \frac{W_1 + \frac{3W}{4}}{A} = \frac{4W_1 + 3W}{4A} \]
Step 5: Conclusion.
The stress at the given point is \(\dfrac{4W_1 + 3W}{4A}\).