Which of the following is unitless quantity?

Options

A. Velocity gradient
B. Pressure gradinet
C. Displecement gradient
D. Force gradient

Answer

To determine which of the following quantities is unitless, we will analyze each option step by step.

### Step 1: Analyze Velocity Gradient
- **Definition**: The velocity gradient is defined as the change in velocity per unit length, expressed mathematically as \( \frac{dv}{dx} \).
- **Units**: The SI unit of velocity is meters per second (m/s). Therefore, the unit of velocity gradient becomes:
\[ \text{Velocity Gradient} = \frac{\text{m/s}}{\text{m}} = \frac{1}{\text{s}} \text{ (per second)} \]
- **Conclusion**: Since the velocity gradient has units (1/s), it is not unitless.

### Step 2: Analyze Pressure Gradient
- **Definition**: The pressure gradient is defined as the change in pressure per unit length, expressed as \( \frac{dp}{dx} \).
- **Units**: The SI unit of pressure is pascal (Pa), which can be expressed as:
\[ \text{Pressure} = \frac{\text{Force}}{\text{Area}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}^2} = \frac{\text{kg}}{\text{m} \cdot \text{s}^2} \]
Therefore, the unit of pressure gradient becomes:
\[ \text{Pressure Gradient} = \frac{\text{Pa}}{\text{m}} = \frac{\text{kg/m/s}^2}{\text{m}} = \frac{\text{kg}}{\text{m}^2 \cdot \text{s}^2} \]
- **Conclusion**: Since the pressure gradient has units (kg/m²·s²), it is not unitless.

### Step 3: Analyze Displacement Gradient
- **Definition**: The displacement gradient is defined as the change in displacement per unit change in position, expressed as \( \frac{ds}{dx} \).
- **Units**: The unit of displacement is meters (m). Thus, the unit of displacement gradient becomes:
\[ \text{Displacement Gradient} = \frac{\text{m}}{\text{m}} = 1 \text{ (unitless)} \]
- **Conclusion**: Since the displacement gradient has no units, it is unitless.

### Step 4: Analyze Force Gradient
- **Definition**: The force gradient is defined as the change in force per unit change in position, expressed as \( \frac{dF}{dx} \).
- **Units**: The SI unit of force is newton (N), which can be expressed as:
\[ \text{Force} = \text{kg} \cdot \text{m/s}^2 \]
Therefore, the unit of force gradient becomes:
\[ \text{Force Gradient} = \frac{\text{N}}{\text{m}} = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}} = \frac{\text{kg}}{\text{s}^2} \]
- **Conclusion**: Since the force gradient has units (kg/s²), it is not unitless.

### Final Conclusion
After analyzing all options, we find that:
- The unitless quantity is **Displacement Gradient** (Option 3).

### Summary
The correct answer is **Option 3: Displacement Gradient**.

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