The dimensional formula of velocity gradient is

Options

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

Answer

To find the dimensional formula of the velocity gradient, we can follow these steps:

### Step 1: Understand the Definition of Velocity Gradient
The velocity gradient is defined as the rate of change of velocity with respect to distance. Mathematically, it can be expressed as:

\[ \text{Velocity Gradient} = \frac{\Delta v}{\Delta l} \]

where \( \Delta v \) is the change in velocity and \( \Delta l \) is the change in distance.

### Step 2: Identify the Dimensions of Velocity and Distance
1. **Velocity (v)**: The dimensional formula for velocity is derived from its definition as distance per unit time. Therefore, the dimensional formula for velocity is:

\[ [v] = \frac{[l]}{[t]} = \frac{L}{T} = M^0 L^1 T^{-1} \]

2. **Distance (l)**: The dimensional formula for distance is simply:

\[ [l] = L = M^0 L^1 T^0 \]

### Step 3: Write the Dimensional Formula for Velocity Gradient
Now, substituting the dimensional formulas into the expression for velocity gradient:

\[ \text{Velocity Gradient} = \frac{[v]}{[l]} = \frac{M^0 L^1 T^{-1}}{M^0 L^1 T^0} \]

### Step 4: Simplify the Expression
When we divide the dimensions, we subtract the exponents of like terms:

\[ \text{Velocity Gradient} = M^{0-0} L^{1-1} T^{-1-0} = M^0 L^0 T^{-1} \]

### Step 5: Final Result
Thus, the dimensional formula for the velocity gradient is:

\[ \text{Velocity Gradient} = M^0 L^0 T^{-1} \]

### Conclusion
The dimensional formula of the velocity gradient is \( M^0 L^0 T^{-1} \).

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