A substance breaks down by a stress of `10^(6) Nm^(-2)`. If the density of the material of the wire is `3 xx 10^(3) kgm^(-3)`. Then the length of the wire of the substance which will break under its own weight when suspended vertically is

Options

A. 66.6 m
B. 60.0 m
C. 33.3 m
D. 30.9 m

Explanation

To solve the problem, we need to determine the length of a wire made from a substance that will break under its own weight when suspended vertically. We are given the breaking stress and the density of the material.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Breaking stress, \( \sigma = 10^6 \, \text{N/m}^2 \)
- Density, \( \rho = 3 \times 10^3 \, \text{kg/m}^3 \)
- Acceleration due to gravity, \( g \approx 10 \, \text{m/s}^2 \)

2. **Understand the Concept of Breaking Stress:**
- Breaking stress is defined as the force per unit area that a material can withstand before failure. Mathematically, it is given by:
\[ \sigma = \frac{F}{A} \]
- In this case, the breaking force \( F \) will be the weight of the wire, which can be expressed as:
\[ F = mg \]
- Here, \( m \) is the mass of the wire, and \( A \) is the cross-sectional area.

3. **Express Mass in Terms of Density:**
- The mass \( m \) of the wire can be expressed in terms of its volume \( V \) and density \( \rho \):
\[ m = \rho V \]
- The volume \( V \) of the wire can be expressed as:
\[ V = A \cdot L \]
- Therefore, the mass can be rewritten as:
\[ m = \rho A L \]

4. **Substituting into the Breaking Stress Equation:**
- Substitute \( m \) into the breaking stress equation:
\[ \sigma = \frac{mg}{A} = \frac{\rho A L g}{A} \]
- This simplifies to:
\[ \sigma = \rho L g \]

5. **Rearranging to Find Length \( L \):**
- Rearranging the equation to solve for \( L \):
\[ L = \frac{\sigma}{\rho g} \]

6. **Substituting the Values:**
- Now substitute the known values into the equation:
\[ L = \frac{10^6 \, \text{N/m}^2}{(3 \times 10^3 \, \text{kg/m}^3)(10 \, \text{m/s}^2)} \]
- Simplifying the denominator:
\[ L = \frac{10^6}{3 \times 10^4} = \frac{10^6}{3 \times 10^4} = \frac{10^2}{3} \approx 33.33 \, \text{m} \]

7. **Final Answer:**
- The length of the wire that will break under its own weight is approximately:
\[ L \approx 33.33 \, \text{m} \]