A rubber cord of density d, Young's modulus Y and length L is suspended vertically . If the cord extends by a length 0.5 L under its own weight , then L is

Options

A. `Y/(2dg)`
B. `Y/(dg)`
C. `(2Y)/(dy)`
D. `(dg)/(2Y)`

Answer

To solve the problem, we need to find the length \( L \) of a rubber cord that extends by a length \( 0.5L \) under its own weight. We will use the concepts of Young's modulus, density, and the relationship between stress and strain.

### Step-by-Step Solution:

1. **Understanding Young's Modulus**:
Young's modulus \( Y \) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
where \( F \) is the force applied, \( A \) is the cross-sectional area, \( \Delta L \) is the change in length, and \( L \) is the original length.

2. **Weight of the Cord**:
The weight of the cord acts downward due to gravity. The force \( F \) acting on a small segment of the cord of length \( x \) is given by:
\[ F = \text{mass} \times g = \rho A x g \]
where \( \rho \) is the density of the cord, \( A \) is the cross-sectional area, and \( g \) is the acceleration due to gravity.

3. **Change in Length**:
The total change in length \( \Delta L \) of the cord can be expressed as:
\[ \Delta L = \frac{F L}{A Y} \]
Here, \( F \) is the weight of the entire cord, which can be expressed as:
\[ F = \rho A L g \]

4. **Substituting for \( \Delta L \)**:
Substituting the expression for \( F \) into the equation for \( \Delta L \):
\[ \Delta L = \frac{\rho A L g \cdot L}{A Y} = \frac{\rho g L^2}{Y} \]

5. **Setting Up the Equation**:
According to the problem, the cord extends by \( 0.5L \):
\[ 0.5L = \frac{\rho g L^2}{Y} \]

6. **Rearranging the Equation**:
Rearranging this equation to solve for \( L \):
\[ L = \frac{0.5 Y}{\rho g} \]

7. **Using Density**:
Since density \( \rho \) can be expressed in terms of \( d \) (the given density), we can substitute:
\[ L = \frac{0.5 Y}{d g} \]

8. **Final Expression**:
Thus, the length \( L \) of the rubber cord is given by:
\[ L = \frac{Y}{2dg} \]

### Conclusion:
The length \( L \) of the rubber cord that extends by \( 0.5L \) under its own weight is:
\[ L = \frac{Y}{2dg} \]