The dimensional formula for surface tension is

Options

A. `[ML^(0)T^(-2)]`
B. `[MLT^(-1)]`
C. `[M^(2)L^(-1)T]`
D. `[M^(0)L^(0)T^(2)]`

Answer

To find the dimensional formula for surface tension, we can follow these steps:

### Step 1: Understand the Definition of Surface Tension
Surface tension (γ) is defined as the force (F) per unit length (L). Mathematically, it is expressed as:
\[ \gamma = \frac{F}{L} \]

### Step 2: Determine the Dimensions of Force
Force (F) is defined by Newton's second law, which states that force is the product of mass (m) and acceleration (a):
\[ F = m \cdot a \]

The dimensional formula for mass (m) is:
\[ [M] \]

The dimensional formula for acceleration (a) is given by:
\[ a = \frac{v}{t} \]
where \( v \) is velocity and \( t \) is time. The dimensional formula for velocity (v) is:
\[ v = \frac{s}{t} \]
where \( s \) is distance. Therefore, the dimensional formula for acceleration becomes:
\[ [a] = \frac{[L][T^{-2}]}{[T]} = [L][T^{-2}] \]

Thus, the dimensional formula for force (F) is:
\[ [F] = [M][L][T^{-2}] \]

### Step 3: Determine the Dimensions of Length
The dimensional formula for length (L) is simply:
\[ [L] \]

### Step 4: Combine the Dimensions for Surface Tension
Now, substituting the dimensions of force and length into the formula for surface tension:
\[ \gamma = \frac{F}{L} \]
The dimensional formula for surface tension becomes:
\[ [\gamma] = \frac{[M][L][T^{-2}]}{[L]} \]

### Step 5: Simplify the Expression
When we simplify the expression, the length (L) in the numerator and denominator cancels out:
\[ [\gamma] = [M][T^{-2}] \]

### Final Answer
Thus, the dimensional formula for surface tension is:
\[ [\gamma] = [M^1][L^0][T^{-2}] \]