The breaking stress for a substance is `10^(6)N//m^(2)`. What length of the wire of this substance should be suspended vertically so that the wire breaks under its own weight? (Given: density of material of the wire `=4xx10^(3)kg//m^(3)` and `g=10 ms^(-2))`

Options

A. 10 m
B. 15 m
C. 25 m
D. 34 m

Explanation

To solve the problem of finding the length of the wire that will break under its own weight, we can follow these steps:

### Step 1: Understand the relationship between stress, tension, and length
The breaking stress (σ) is defined as the maximum stress that a material can withstand before failure. It is given by the formula:
\[ \sigma = \frac{T}{A} \]
where \(T\) is the tension in the wire and \(A\) is the cross-sectional area of the wire.

### Step 2: Determine the tension in the wire
For a wire suspended vertically, the tension at any point is due to the weight of the wire below that point. The maximum tension (which occurs at the top of the wire) can be expressed as:
\[ T = \text{mass} \times g \]
The mass of the wire can be calculated as:
\[ \text{mass} = \text{density} \times \text{volume} = \rho \times A \times L \]
where \(L\) is the length of the wire, \(\rho\) is the density, and \(A\) is the cross-sectional area.

### Step 3: Substitute the mass into the tension equation
Substituting the mass into the tension equation gives:
\[ T = \rho \times A \times L \times g \]

### Step 4: Relate tension to stress
From the stress formula, we have:
\[ \sigma = \frac{T}{A} \]
Substituting the expression for \(T\) gives:
\[ \sigma = \frac{\rho \times A \times L \times g}{A} \]
This simplifies to:
\[ \sigma = \rho \times L \times g \]

### Step 5: Rearrange the equation to solve for \(L\)
To find the length \(L\), we can rearrange the equation:
\[ L = \frac{\sigma}{\rho \times g} \]

### Step 6: Substitute the known values
Given:
- Breaking stress, \(\sigma = 10^6 \, \text{N/m}^2\)
- Density, \(\rho = 4 \times 10^3 \, \text{kg/m}^3\)
- Gravitational acceleration, \(g = 10 \, \text{m/s}^2\)

Substituting these values into the equation:
\[ L = \frac{10^6}{4 \times 10^3 \times 10} \]

### Step 7: Calculate \(L\)
Calculating the denominator:
\[ 4 \times 10^3 \times 10 = 4 \times 10^4 \]
Now substituting back:
\[ L = \frac{10^6}{4 \times 10^4} = \frac{10^6}{4 \times 10^4} = \frac{10^2}{4} = 25 \, \text{m} \]

### Final Answer
The length of the wire that should be suspended vertically so that it breaks under its own weight is:
\[ \boxed{25 \, \text{m}} \]