A BREAKING STRESS OF THE MATERIAL OF A WIRE IS 10⁶ N/M² . THE MAXIMUM LENGTH OF THE WIRE FOR WHICH IT BREAKS UNDER ITS OWN WEIGHT WILL BE (DENSITY OF MATERIAL IS 2 X 10³ KG/M³)?

To determine the maximum length of a wire that can hang under its own weight before breaking, we can use the concept of tensile stress and the relationship between stress, force, and area. The breaking stress of the material gives us a threshold that we cannot exceed, while the weight of the wire itself creates a force that increases with length. Let's break this down step by step.

Understanding the Problem

The breaking stress of the wire is given as 10⁶ N/m², and the density of the material is 2 x 10³ kg/m³. We need to find the maximum length of the wire, denoted as L, that can hang without breaking due to its own weight.

Key Concepts

• Tensile Stress: This is defined as the force (F) applied per unit area (A) of the wire. It can be expressed as:
  Stress = F/A
• Weight of the Wire: The weight (W) of the wire can be calculated using the formula:
  W = mass x gravity = (density x volume) x g
• Volume of the Wire: For a wire of length L and cross-sectional area A, the volume (V) is:
  V = A x L

Calculating the Maximum Length

1. **Calculate the weight of the wire:**
The mass of the wire can be expressed as:
mass = density x volume = density x (A x L)
Therefore, the weight of the wire is:
W = density x (A x L) x g
Substituting the density (2 x 10³ kg/m³) and using g = 9.81 m/s²:
W = (2 x 10³) x (A x L) x 9.81

2. **Relate the weight to tensile stress:**
The tensile stress at the breaking point is given as 10⁶ N/m². The force acting on the wire due to its weight is equal to the tensile stress multiplied by the cross-sectional area:
Stress = W/A
Therefore, we can set up the equation:
10⁶ = (density x A x L x g) / A
Simplifying this gives us:
10⁶ = (2 x 10³ x L x 9.81)

3. **Solve for L:**
Rearranging the equation to find L:
L = 10⁶ / (2 x 10³ x 9.81)
Now, calculating the values:
L = 10⁶ / (19620) ≈ 51.0 m

Final Thoughts

The maximum length of the wire that can hang under its own weight without breaking is approximately 51.0 meters. This calculation illustrates how material properties, such as density and breaking stress, directly influence the structural limits of materials in practical applications. Understanding these principles is crucial in fields like engineering and materials science, where safety and reliability are paramount.