Expermients show that frequency (n) of a tuning fork depends on lentght (I) fo the prong, density (d) and the Young's modulus (Y) of its meterial. On the basis of dimensional analysis, dericve an expression for frequency of tunnig fork.

Answer

To derive an expression for the frequency (n) of a tuning fork based on the length (L) of its prong, density (d), and Young's modulus (Y) of its material using dimensional analysis, we can follow these steps:

### Step 1: Identify the Variables
We know that the frequency \( n \) depends on:
- Length of the prong \( L \)
- Density of the material \( d \)
- Young's modulus of the material \( Y \)

### Step 2: Write the Relationship
We can express the relationship as:
\[ n = k \cdot L^A \cdot d^B \cdot Y^C \]
where \( k \) is a dimensionless constant, and \( A \), \( B \), and \( C \) are the powers we need to determine.

### Step 3: Write the Dimensions
Next, we need to write the dimensions of each variable:
- Frequency \( n \) has dimensions of \( [T^{-1}] \)
- Length \( L \) has dimensions of \( [L] \)
- Density \( d \) has dimensions of \( [M L^{-3}] \)
- Young's modulus \( Y \) has dimensions of \( [M L^{-1} T^{-2}] \)

### Step 4: Substitute the Dimensions
Substituting the dimensions into the equation gives:
\[ [T^{-1}] = [L]^A \cdot [M L^{-3}]^B \cdot [M L^{-1} T^{-2}]^C \]

Expanding this, we have:
\[ [T^{-1}] = [L]^A \cdot [M]^B \cdot [L]^{-3B} \cdot [M]^C \cdot [L]^{-C} \cdot [T]^{-2C} \]

### Step 5: Combine the Dimensions
Combining the dimensions on the right side:
\[ [T^{-1}] = [M]^{B+C} \cdot [L]^{A - 3B - C} \cdot [T]^{-2C} \]

### Step 6: Set Up the Equations
Now, we equate the powers of \( M \), \( L \), and \( T \) from both sides:
1. For mass \( M \): \( B + C = 0 \) (1)
2. For length \( L \): \( A - 3B - C = 0 \) (2)
3. For time \( T \): \( -2C = -1 \) (3)

### Step 7: Solve the Equations
From equation (3):
\[ C = \frac{1}{2} \]

Substituting \( C \) into equation (1):
\[ B + \frac{1}{2} = 0 \]
\[ B = -\frac{1}{2} \]

Now substitute \( B \) and \( C \) into equation (2):
\[ A - 3(-\frac{1}{2}) - \frac{1}{2} = 0 \]
\[ A + \frac{3}{2} - \frac{1}{2} = 0 \]
\[ A + 1 = 0 \]
\[ A = -1 \]

### Step 8: Write the Final Expression
Now we have:
- \( A = -1 \)
- \( B = -\frac{1}{2} \)
- \( C = \frac{1}{2} \)

Substituting these values back into the original equation:
\[ n = k \cdot L^{-1} \cdot d^{-\frac{1}{2}} \cdot Y^{\frac{1}{2}} \]

Rearranging gives:
\[ n = k \cdot \frac{1}{L} \cdot \sqrt{\frac{Y}{d}} \]

### Final Expression
Thus, the expression for the frequency \( n \) of the tuning fork is:
\[ n = k \cdot \frac{1}{L} \cdot \sqrt{\frac{Y}{d}} \]