A point mass m is suspended at the end of a massless wire of length l and cross section. If Y is the Young's modulus for the wire, obtain the frequency of oscillation for the simple harmonic motion along the vertical line.

Options

A. `(1)/(2pi)sqrt((LA)/(mY))`
B. `(1)/(2pi)sqrt((LAm)/(Y))`
C. `(1)/(2pi)sqrt((YA)/(mL))`
D. `(1)/(2pi)sqrt((mY)/(AL))`

Explanation

To find the frequency of oscillation for a point mass \( m \) suspended at the end of a massless wire of length \( l \) and cross-sectional area \( A \), with Young's modulus \( Y \), we can follow these steps:

### Step 1: Understand the System
We have a mass \( m \) hanging from a wire. When the mass is displaced from its equilibrium position and released, it will undergo simple harmonic motion (SHM).

### Step 2: Apply Young's Modulus
Young's modulus \( Y \) relates stress and strain in the wire. The formula for Young's modulus is given by:

\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta l/l} \]

Where:
- \( F \) is the force applied (weight of the mass, \( mg \))
- \( A \) is the cross-sectional area of the wire
- \( \Delta l \) is the change in length of the wire
- \( l \) is the original length of the wire

Rearranging this gives us:

\[ F = Y \cdot A \cdot \frac{\Delta l}{l} \]

### Step 3: Relate Force to Displacement
In SHM, the restoring force \( F \) can also be expressed as:

\[ F = -k \Delta x \]

Where \( k \) is the spring constant and \( \Delta x \) is the displacement from the equilibrium position. By comparing both expressions for force, we can equate them:

\[ mg = Y \cdot A \cdot \frac{\Delta l}{l} \]

### Step 4: Identify the Spring Constant
From the expression \( F = Y \cdot A \cdot \frac{\Delta l}{l} \), we can identify the spring constant \( k \):

\[ k = \frac{Y \cdot A}{l} \]

### Step 5: Determine the Frequency of Oscillation
The frequency \( f \) of oscillation for a mass-spring system is given by:

\[ f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} \]

Substituting the expression for \( k \):

\[ f = \frac{1}{2\pi} \sqrt{\frac{Y \cdot A / l}{m}} \]

### Final Expression
Thus, the frequency of oscillation is:

\[ f = \frac{1}{2\pi} \sqrt{\frac{Y \cdot A}{m \cdot l}} \]

### Summary
The frequency of oscillation for the simple harmonic motion of the mass suspended from the wire is:

\[ f = \frac{1}{2\pi} \sqrt{\frac{Y \cdot A}{m \cdot l}} \]