Frequency is the function of density `(rho)` , length `(a)` and surface tension `(T)` . Then its value is

Options

A. `krho^(1//2) l^(3//2) T^(-1//2)`
B. `krho^(3//2) l^(3//2) T^(-1//2)`
C. `krho^(1//2) l^(3//2) T^(-3//4)`
D. `kT^(1//2) rho^(-1//2) l^(-3//2)`

Answer

To find the relationship for frequency (f) as a function of density (ρ), length (a), and surface tension (T), we can use dimensional analysis. Here’s a step-by-step solution:

### Step 1: Define the relationship
Assume that frequency (f) can be expressed as:
\[ f = K \cdot \rho^A \cdot a^B \cdot T^C \]
where \( K \) is a dimensionless constant, and \( A \), \( B \), and \( C \) are the powers we need to determine.

### Step 2: Write down the dimensions
The dimensions of frequency (f) are:
\[ [f] = T^{-1} \]

Now, we will write the dimensions of each variable:
- Density (ρ):
\[ [\rho] = \frac{M}{L^3} = M L^{-3} \]
- Length (a):
\[ [a] = L \]
- Surface tension (T):
\[ [T] = \frac{F}{L} = \frac{M L T^{-2}}{L} = M T^{-2} \]

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

### Step 4: Expand the right-hand side
This expands to:
\[ T^{-1} = M^{A+C} \cdot L^{-3A + B} \cdot T^{-2C} \]

### Step 5: Equate the dimensions
Now we equate the dimensions on both sides:
1. For mass (M):
\[ A + C = 0 \]
2. For length (L):
\[ -3A + B = 0 \]
3. For time (T):
\[ -2C = -1 \]

### Step 6: Solve the equations
From the third equation:
\[ -2C = -1 \implies C = \frac{1}{2} \]

Substituting \( C \) into the first equation:
\[ A + \frac{1}{2} = 0 \implies A = -\frac{1}{2} \]

Now substituting \( A \) into the second equation:
\[ -3(-\frac{1}{2}) + B = 0 \implies \frac{3}{2} + B = 0 \implies B = -\frac{3}{2} \]

### Step 7: Write the final expression
Now we can substitute the values of \( A \), \( B \), and \( C \) back into the original equation:
\[ f = K \cdot \rho^{-\frac{1}{2}} \cdot a^{-\frac{3}{2}} \cdot T^{\frac{1}{2}} \]

### Step 8: Simplify the expression
This can be rewritten as:
\[ f = K \cdot \frac{T^{\frac{1}{2}}}{\rho^{\frac{1}{2}} \cdot a^{\frac{3}{2}}} \]

### Conclusion
Thus, the frequency is given by:
\[ f \propto \frac{T^{\frac{1}{2}}}{\rho^{\frac{1}{2}} \cdot a^{\frac{3}{2}}} \]