The fundamental quantity which has the same power in the dimensional formula of surface tension and coefficient of viscosity is

Options

A. Mass
B. Length
C. Time
D. None

Answer

To solve the problem of identifying the fundamental quantity that has the same power in the dimensional formulas of surface tension and coefficient of viscosity, we will follow these steps:

### Step 1: Understand Surface Tension
Surface tension (σ) is defined as the force (F) per unit length (L). The formula can be expressed as:
\[ \sigma = \frac{F}{L} \]

### Step 2: Determine the Dimensions of Surface Tension
The dimension of force (F) is given by:
\[ [F] = [M][L][T^{-2}] \]
where M is mass, L is length, and T is time.

The dimension of length (L) is simply:
\[ [L] = [L] \]

Thus, the dimension of surface tension (σ) becomes:
\[ [\sigma] = \frac{[F]}{[L]} = \frac{[M][L][T^{-2}]}{[L]} = [M][L^0][T^{-2}] \]
So, the dimensional formula for surface tension is:
\[ [\sigma] = [M][T^{-2}] \]

### Step 3: Understand Coefficient of Viscosity
The coefficient of viscosity (η) can be defined from the relationship between force, area (A), and the velocity gradient (dv/dx):
\[ F = \eta A \left(\frac{dv}{dx}\right) \]

### Step 4: Determine the Dimensions of Coefficient of Viscosity
The dimension of area (A) is:
\[ [A] = [L^2] \]

The dimension of the velocity gradient (dv/dx) can be expressed as:
\[ \frac{dv}{dx} = \frac{[L][T^{-1}]}{[L]} = [T^{-1}] \]

Substituting these into the equation for force gives:
\[ [F] = [\eta][L^2][T^{-1}] \]

Rearranging for the coefficient of viscosity (η):
\[ [\eta] = \frac{[F]}{[A][dv/dx]} = \frac{[M][L][T^{-2}]}{[L^2][T^{-1}]} = [M][L^{-1}][T^{-1}] \]

### Step 5: Compare the Dimensions
Now we have the dimensional formulas:
- For surface tension:
\[ [\sigma] = [M][T^{-2}] \]
- For coefficient of viscosity:
\[ [\eta] = [M][L^{-1}][T^{-1}] \]

### Step 6: Identify the Common Fundamental Quantity
From the dimensional formulas, we can see that the fundamental quantity that has the same power in both formulas is mass (M), as it appears with a power of 1 in both cases.

### Final Answer
The fundamental quantity which has the same power in the dimensional formula of surface tension and coefficient of viscosity is:
**Mass (M)**