Which one of the following represents the correct dimensions of the coefficient of viscosity?

Options

A. `[ML ^-1T^-2]`
B. `[MLT^-1]`
C. `[ML^-1T^-1]`
D. `[ML^-2T^-2]`

Answer

To determine the correct dimensions of the coefficient of viscosity (η), we can follow these steps:

### Step 1: Understand the definition of viscosity
The coefficient of viscosity (η) is defined in the context of the force acting on a fluid due to its viscosity. It can be derived from the equation for viscous force.

### Step 2: Write the formula for viscous force
The viscous force (F) acting on a cylindrical object moving through a fluid can be expressed as:
\[ F = 6 \pi \eta R V \]
where:
- \( F \) = viscous force
- \( \eta \) = coefficient of viscosity
- \( R \) = radius of the cylindrical object
- \( V \) = velocity of the object

### Step 3: Rearrange the formula to solve for η
From the equation, we can rearrange it to find the coefficient of viscosity:
\[ \eta = \frac{F}{6 \pi R V} \]
Since \( 6 \pi \) is a dimensionless constant, we can ignore it for dimensional analysis.

### Step 4: Identify the dimensions of each variable
- The dimension of force \( F \) is given by Newton (N), which can be expressed in terms of base dimensions:
\[ [F] = [M][L][T^{-2}] \]
- The dimension of radius \( R \) is:
\[ [R] = [L] \]
- The dimension of velocity \( V \) is:
\[ [V] = \frac{[L]}{[T]} \]

### Step 5: Substitute the dimensions into the equation for η
Now we can substitute the dimensions into the equation for η:
\[ [\eta] = \frac{[F]}{[R][V]} = \frac{[M][L][T^{-2}]}{[L] \cdot \frac{[L]}{[T]}} \]

### Step 6: Simplify the expression
When we simplify this expression:
\[ [\eta] = \frac{[M][L][T^{-2}]}{[L^2][T^{-1}]} = \frac{[M][L][T^{-2}]}{[L^2]} \cdot [T] \]
\[ [\eta] = [M][L^{-1}][T^{-1}] \]

### Conclusion
Thus, the dimensions of the coefficient of viscosity (η) are:
\[ [\eta] = [M][L^{-1}][T^{-1}] \]

### Final Answer
The correct representation of the dimensions of the coefficient of viscosity is:
\[ \text{Option C: } [M][L^{-1}][T^{-1}] \]

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