Which of the following quantity is unitless ?

Options

A. Velocity gradient
B. Pressure gradient
C. Displacement gradient
D. Force gradient

Answer

To determine which of the following quantities is unitless, we will analyze each option provided: velocity gradient, pressure gradient, displacement gradient, and force gradient.

### Step-by-Step Solution:

1. **Understanding Gradient**:
- A gradient refers to the rate of change of a quantity with respect to displacement. Mathematically, it can be represented as the change in a quantity (ΔA) divided by the change in displacement (Δx), or dA/dx.

2. **Analyzing Each Option**:
- **Velocity Gradient**:
- Definition: Rate of change of velocity with respect to displacement.
- Formula: \( \text{Velocity Gradient} = \frac{dv}{dx} \)
- Units: Velocity (m/s) divided by displacement (m) gives us \( \frac{m/s}{m} = \frac{1}{s} \). So, it has units (1/s).

- **Pressure Gradient**:
- Definition: Rate of change of pressure with respect to displacement.
- Formula: \( \text{Pressure Gradient} = \frac{dP}{dx} \)
- Units: Pressure (N/m² or kg/(m·s²)) divided by displacement (m) gives us \( \frac{kg/(m·s²)}{m} = \frac{kg}{m²·s²} \). So, it has units (kg/(m²·s²)).

- **Displacement Gradient**:
- Definition: Rate of change of displacement with respect to displacement.
- Formula: \( \text{Displacement Gradient} = \frac{dx}{dx} \)
- Units: Displacement (m) divided by displacement (m) gives us \( \frac{m}{m} = 1 \). So, it is unitless.

- **Force Gradient**:
- Definition: Rate of change of force with respect to displacement.
- Formula: \( \text{Force Gradient} = \frac{dF}{dx} \)
- Units: Force (N or kg·m/s²) divided by displacement (m) gives us \( \frac{kg·m/s²}{m} = \frac{kg}{s²} \). So, it has units (kg/s²).

3. **Conclusion**:
- Among the given options, only the displacement gradient is unitless. Therefore, the correct answer is **C) Displacement Gradient**.