When a liquid flows in a tube, there is relative motion between adjacent layers of the liquid. This force is called the viscous force which tends to oppose the relative motion between the layers of the liquid. Newton was the first person to study the factors that govern the viscous force in a liquid. According to Newton’s law of viscous flow, the magnitude of the viscous force on a certain layer of a liquid is given by
` F = - eta A (dv)/(dx)`
where A is the area of the layer ` (dv)/(dx) ` is the velocity gradient at the layer and ` eta ` is the coefficient of viscosity of the liquid.
The dimensional formula for the coefficient of viscosity is :

Options

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

Explanation

To find the dimensional formula for the coefficient of viscosity (η), we start from the equation given in the question:

\[ F = -\eta A \frac{dv}{dx} \]

Where:
- \( F \) is the viscous force,
- \( \eta \) is the coefficient of viscosity,
- \( A \) is the area,
- \( \frac{dv}{dx} \) is the velocity gradient.

### Step 1: Rearranging the equation
We can rearrange the equation to solve for η:

\[ \eta = \frac{F}{A \frac{dv}{dx}} \]

### Step 2: Finding the dimensions of each term

1. **Dimension of Force (F)**:
- Force is given by \( F = ma \) (mass × acceleration).
- The dimensional formula for mass (m) is \( [M] \).
- Acceleration (a) has the dimensional formula \( [L T^{-2}] \).
- Therefore, the dimension of force is:
\[ [F] = [M][L T^{-2}] = [M L T^{-2}] \]

2. **Dimension of Area (A)**:
- Area is given by \( A = l^2 \) (length × length).
- The dimensional formula for length (l) is \( [L] \).
- Therefore, the dimension of area is:
\[ [A] = [L^2] \]

3. **Dimension of Velocity (v)**:
- Velocity is given by \( v = \frac{l}{t} \) (length/time).
- Therefore, the dimension of velocity is:
\[ [v] = [L T^{-1}] \]

4. **Dimension of Velocity Gradient (\( \frac{dv}{dx} \))**:
- The velocity gradient \( \frac{dv}{dx} \) has the dimension of velocity divided by length.
- Therefore, the dimension of the velocity gradient is:
\[ \left[\frac{dv}{dx}\right] = \frac{[L T^{-1}]}{[L]} = [T^{-1}] \]

### Step 3: Substituting the dimensions into the equation for η
Now, substituting the dimensions back into the equation for η:

\[ \eta = \frac{[M L T^{-2}]}{[L^2][T^{-1}]} \]

### Step 4: Simplifying the expression
Now, simplify the expression:

\[ \eta = \frac{[M L T^{-2}]}{[L^2][T^{-1}]} = \frac{[M L T^{-2}]}{[L^2]} \cdot [T] = [M L^{-1} T^{-1}] \]

### Final Result
Thus, the dimensional formula for the coefficient of viscosity (η) is:

\[ \eta = [M L^{-1} T^{-1}] \]