Which one of the following is dimensionless physical quantity?

Options

A. Velocity gradient
B. Stress
C. Force gradient
D. Angle

Answer

To determine which of the given options is a dimensionless physical quantity, we will analyze each option step by step.

### Step 1: Understand Dimensionless Physical Quantity
A dimensionless physical quantity is one that has no dimensions, represented as \( M^0 L^0 T^0 \). This means it does not depend on mass (M), length (L), or time (T).

### Step 2: Analyze Each Option

#### Option A: Velocity Gradient
- **Definition**: Velocity gradient is defined as the change in velocity per unit length.
- **Formula**:
\[ \text{Velocity Gradient} = \frac{\text{Velocity}}{\text{Length}} \]
- **Dimensions**:
- Velocity has dimensions \( [L T^{-1}] \).
- Length has dimensions \( [L] \).
- **Calculation**:
\[ \text{Dimensions of Velocity Gradient} = \frac{[L T^{-1}]}{[L]} = [T^{-1}] \]
- **Conclusion**: Since it has dimensions \( [T^{-1}] \), it is not dimensionless.

#### Option B: Stress
- **Definition**: Stress is defined as force per unit area.
- **Formula**:
\[ \text{Stress} = \frac{\text{Force}}{\text{Area}} \]
- **Dimensions**:
- Force has dimensions \( [M L T^{-2}] \).
- Area has dimensions \( [L^2] \).
- **Calculation**:
\[ \text{Dimensions of Stress} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]
- **Conclusion**: Since it has dimensions \( [M L^{-1} T^{-2}] \), it is not dimensionless.

#### Option C: Force Gradient
- **Definition**: Force gradient is defined as the change in force per unit length.
- **Formula**:
\[ \text{Force Gradient} = \frac{\text{Force}}{\text{Length}} \]
- **Dimensions**:
- Force has dimensions \( [M L T^{-2}] \).
- Length has dimensions \( [L] \).
- **Calculation**:
\[ \text{Dimensions of Force Gradient} = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]
- **Conclusion**: Since it has dimensions \( [M T^{-2}] \), it is not dimensionless.

#### Option D: Angle
- **Definition**: Angle is defined as the ratio of arc length to radius.
- **Formula**:
\[ \text{Angle} = \frac{\text{Arc Length}}{\text{Radius}} \]
- **Dimensions**:
- Arc length has dimensions \( [L] \).
- Radius has dimensions \( [L] \).
- **Calculation**:
\[ \text{Dimensions of Angle} = \frac{[L]}{[L]} = [1] = [M^0 L^0 T^0] \]
- **Conclusion**: Since it has no dimensions, it is dimensionless.

### Final Answer
The dimensionless physical quantity among the options is **Angle**.

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