Which of the following is unitless quantity?

Options

A. Pressure gradient
B. Displacement gradient
C. force gradient
D. velocity gradient.

Explanation

To determine which of the following is a unitless quantity among pressure gradient, displacement gradient, force gradient, and velocity gradient, we can follow these steps:

### Step 1: Understand the concept of a gradient
A gradient of a quantity is defined as the change in that quantity (ΔQ) divided by the change in position (Δx). Mathematically, it can be expressed as:
\[ \text{Gradient} = \frac{\Delta Q}{\Delta x} \]

### Step 2: Identify the quantities involved
We have the following quantities:
1. Pressure (P)
2. Displacement (x)
3. Force (F)
4. Velocity (V)

### Step 3: Write the gradient expressions for each quantity
Using the formula for gradient, we can write:
1. Pressure gradient: \( \frac{\Delta P}{\Delta x} \)
2. Displacement gradient: \( \frac{\Delta x}{\Delta x} \)
3. Force gradient: \( \frac{\Delta F}{\Delta x} \)
4. Velocity gradient: \( \frac{\Delta V}{\Delta x} \)

### Step 4: Analyze the dimensions of each gradient
- For the **pressure gradient**: The units of pressure (P) are force per unit area (e.g., N/m²). Thus, the gradient will have units.
- For the **displacement gradient**: Since both the numerator and denominator are displacement (x), it simplifies to:
\[ \frac{\Delta x}{\Delta x} = 1 \]
This means the displacement gradient is unitless.
- For the **force gradient**: The units of force (F) are Newtons (N). Thus, the gradient will have units.
- For the **velocity gradient**: The units of velocity (V) are meters per second (m/s). Thus, the gradient will have units.

### Step 5: Conclusion
From the analysis, we can conclude that the only unitless quantity among the options provided is the **displacement gradient**.

### Final Answer:
The displacement gradient is the unitless quantity.

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