Units of coefficient of viscosity are

Options

A. `ML^(2)T^(-2)`
B. `ML^(2)T^(-1)`
C. `ML^(-1)T^(-1)`
D. `MLT`

Answer

To find the units of the coefficient of viscosity (η), we can start from the formula that relates force (F), viscosity (η), radius (r), and velocity (v).

### Step-by-Step Solution:

1. **Understand the Formula**:
The formula we are using is:
\[ F = 6 \pi \eta r V \]
Here, F is the force, η is the coefficient of viscosity, r is the radius, and V is the velocity.

2. **Rearranging the Formula**:
To find the coefficient of viscosity, we rearrange the formula:
\[ \eta = \frac{F}{6 \pi r V} \]
Since \(6 \pi\) is a constant and dimensionless, we can ignore it for dimensional analysis:
\[ \eta = \frac{F}{r V} \]

3. **Identify the Dimensions**:
- The dimension of force (F) is given by:
\[ [F] = M L T^{-2} \]
- The dimension of radius (r) is:
\[ [r] = L \]
- The dimension of velocity (V) is:
\[ [V] = L T^{-1} \]

4. **Substituting the Dimensions**:
Now, substituting these dimensions into the equation for η:
\[ [\eta] = \frac{[F]}{[r][V]} = \frac{M L T^{-2}}{L \cdot (L T^{-1})} \]

5. **Simplifying the Expression**:
- The denominator becomes:
\[ [r][V] = L \cdot (L T^{-1}) = L^2 T^{-1} \]
- Now substituting back:
\[ [\eta] = \frac{M L T^{-2}}{L^2 T^{-1}} = \frac{M}{L T^{-1}} = M L^{-1} T^{-1} \]

6. **Final Result**:
Therefore, the units of the coefficient of viscosity (η) are:
\[ [\eta] = M L^{-1} T^{-1} \]

### Conclusion:
The units of the coefficient of viscosity are \(M L^{-1} T^{-1}\).

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