The dimensional formula of coefficient of kinematic viscosity is

Options

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

Answer

To find the dimensional formula of the coefficient of kinematic viscosity, we can follow these steps:

### Step 1: Understand the relationship between kinematic viscosity, dynamic viscosity, and density.
Kinematic viscosity (ν) is defined as the ratio of dynamic viscosity (η) to density (ρ):
\[ \nu = \frac{\eta}{\rho} \]

### Step 2: Write the dimensional formula for density (ρ).
Density is defined as mass per unit volume. The dimensional formula for mass is \( [M] \) and for volume is \( [L^3] \). Therefore, the dimensional formula for density is:
\[ \rho = \frac{[M]}{[L^3]} = [M L^{-3}] \]

### Step 3: Write the dimensional formula for dynamic viscosity (η).
Dynamic viscosity can be expressed in terms of force, distance, area, and velocity. The formula for dynamic viscosity is:
\[ \eta = \frac{F \cdot d}{A \cdot v} \]
Where:
- \( F \) (force) has the dimensional formula \( [M L T^{-2}] \)
- \( d \) (distance) has the dimensional formula \( [L] \)
- \( A \) (area) has the dimensional formula \( [L^2] \)
- \( v \) (velocity) has the dimensional formula \( [L T^{-1}] \)

Now substituting these into the equation for dynamic viscosity:
\[ \eta = \frac{[M L T^{-2}] \cdot [L]}{[L^2] \cdot [L T^{-1}]} \]
Simplifying this gives:
\[ \eta = \frac{[M L^2 T^{-2}]}{[L^3 T^{-1}]} = [M L^{-1} T^{-1}] \]

### Step 4: Substitute the dimensional formulas into the kinematic viscosity formula.
Now we can substitute the dimensional formulas for dynamic viscosity and density into the kinematic viscosity formula:
\[ \nu = \frac{\eta}{\rho} = \frac{[M L^{-1} T^{-1}]}{[M L^{-3}]} \]
This simplifies to:
\[ \nu = [M L^{-1} T^{-1}] \cdot [M^{-1} L^{3}] = [M^{0} L^{2} T^{-1}] \]

### Step 5: Conclusion
Thus, the dimensional formula for the coefficient of kinematic viscosity is:
\[ \nu = [M^{0} L^{2} T^{-1}] \]

### Final Answer
The dimensional formula of the coefficient of kinematic viscosity is \( [M^{0} L^{2} T^{-1}] \).

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