Speed of transverse wave on a straight wire (mass m, length l , area of cross-section A) is. If the young’s modulus of wire is Y, the extension of wire over its natural length, is independent of

Options

A. `l`
B. `m`
C. `A`
D. `Y`

Explanation

To solve the problem regarding the extension of a wire and its independence from certain parameters, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Problem**: We need to find out how the extension of a wire (ΔL) is related to its physical properties and which factors it is independent of.

2. **Use Hooke's Law**: According to Hooke's Law, the stress (σ) in a material is proportional to the strain (ε):
\[ \sigma = Y \cdot \epsilon \]
where \(Y\) is Young's modulus, \(σ = \frac{F}{A}\) (stress), and \(ε = \frac{\Delta L}{L}\) (strain).

3. **Express ΔL**: Rearranging the equation gives:
\[ \frac{F}{A} = Y \cdot \frac{\Delta L}{L} \]
From this, we can express the change in length (ΔL):
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]
Let’s denote this as Equation (1).

4. **Determine the Speed of the Transverse Wave**: The speed of a transverse wave (u) on a wire is given by:
\[ u = \sqrt{\frac{T}{\mu}} \]
where \(T\) is the tension in the wire (which is equal to the force \(F\)) and \(\mu\) is the mass per unit length of the wire, given by:
\[ \mu = \frac{m}{L} \]
Substituting this into the wave speed equation gives:
\[ u = \sqrt{\frac{F}{\frac{m}{L}}} = \sqrt{\frac{F \cdot L}{m}} \]
Let’s denote this as Equation (2).

5. **Combine Equations**: From Equation (1), we have:
\[ \Delta L = \frac{F \cdot L}{A \cdot Y} \]
And from Equation (2):
\[ F = m \cdot u^2 / L \]
Substituting this expression for \(F\) into the equation for ΔL gives:
\[ \Delta L = \frac{\left(\frac{m \cdot u^2}{L}\right) \cdot L}{A \cdot Y} = \frac{m \cdot u^2}{A \cdot Y} \]

6. **Analyze Independence**: From the final expression for ΔL, we can see that it depends on \(m\), \(u\), \(A\), and \(Y\), but it does not depend on \(L\). Therefore, the extension of the wire over its natural length is independent of the natural length \(L\).

### Conclusion:
The extension of the wire over its natural length is independent of its natural length \(L\).