A uniform wire of radius r and length L hanging vertically and fixed at its upper ends, is attached to a weight at its lower end. If Young's modulus for the material of the wire is Y, the extension is

Options

A. inversely proportional to Y.
B. inversely proportional to `r^(2)`.
C. directly proportional to L.
D. inversely proportional to r.

Answer

To solve the problem of finding the extension (ΔL) of a uniform wire hanging vertically with a weight at its lower end, we can follow these steps:

### Step 1: Understand the relationship defined by Young's Modulus
Young's modulus (Y) is defined as the ratio of stress to strain. Mathematically, it can be expressed as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

### Step 2: Define Stress and Strain
- **Stress** is defined as the force (F) applied per unit area (A):
\[ \text{Stress} = \frac{F}{A} \]
- **Strain** is defined as the change in length (ΔL) divided by the original length (L):
\[ \text{Strain} = \frac{\Delta L}{L} \]

### Step 3: Substitute definitions into Young's Modulus
Substituting the definitions of stress and strain into the equation for Young's modulus, we get:
\[ Y = \frac{F/A}{\Delta L/L} \]

### Step 4: Rearrange the equation
Rearranging the equation gives:
\[ Y = \frac{F \cdot L}{A \cdot \Delta L} \]
From this, we can solve for ΔL:
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]

### Step 5: Calculate the Area of the wire
The area (A) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
where r is the radius of the wire.

### Step 6: Substitute the area back into the ΔL equation
Substituting the area back into the equation for ΔL gives:
\[ \Delta L = \frac{F \cdot L}{\pi r^2 \cdot Y} \]

### Step 7: Analyze the relationship
From the final equation, we can see how ΔL is related to other variables:
- ΔL is directly proportional to the force (F) applied.
- ΔL is directly proportional to the original length (L) of the wire.
- ΔL is inversely proportional to the cross-sectional area (πr²) and Young's modulus (Y).

### Final Expression
Thus, the expression for the extension of the wire is:
\[ \Delta L = \frac{F \cdot L}{\pi r^2 \cdot Y} \]