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 step by step, we will derive the expression for the length \( L \) of the rubber cord that extends by \( 0.5L \) under its own weight.

### Step 1: Understand the Problem
We have a rubber cord of density \( d \), Young's modulus \( Y \), and length \( L \) suspended vertically. The cord extends by a length \( 0.5L \) under its own weight. We need to find the value of \( L \).

### Step 2: Define the Weight of the Cord
The weight of the cord can be expressed as:
\[ W = mg \]
where \( m \) is the mass of the cord and \( g \) is the acceleration due to gravity. The mass \( m \) of the cord can be calculated as:
\[ m = \text{density} \times \text{volume} = d \times (A \times L) \]
where \( A \) is the cross-sectional area of the cord.

### Step 3: Substitute the Mass into the Weight Equation
Substituting the expression for mass into the weight equation gives:
\[ W = d \cdot (A \cdot L) \cdot g = dAgL \]

### Step 4: Use Young's Modulus to Relate Extension to Force
According to Young's modulus, the extension \( \Delta L \) of the cord due to the force \( F \) (which is the weight of the cord) is given by:
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]
In our case, the force \( F \) is equal to the weight \( W \):
\[ \Delta L = \frac{dAgL \cdot L}{A \cdot Y} \]

### Step 5: Simplify the Equation
The area \( A \) cancels out:
\[ \Delta L = \frac{d g L^2}{Y} \]

### Step 6: Set the Extension Equal to \( 0.5L \)
We know that the extension \( \Delta L \) is equal to \( 0.5L \):
\[ 0.5L = \frac{d g L^2}{Y} \]

### Step 7: Rearranging the Equation
Rearranging the equation gives:
\[ L = \frac{2Y}{dg} \]

### Final Result
Thus, the length \( L \) of the rubber cord is:
\[ L = \frac{2Y}{dg} \]