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 the concept of dimensionless quantities
A dimensionless quantity is one that has no physical dimensions associated with it. This means that when we express it in terms of the fundamental dimensions (mass, length, time, etc.), all the exponents in the dimensional formula must be zero.

### Step 2: Analyze the first option - Velocity Gradient
- The velocity gradient is defined as the change in velocity per unit length.
- Mathematically, it can be expressed as:
\[ \text{Velocity Gradient} = \frac{\text{Velocity}}{\text{Length}} \]
- The dimension of velocity is \( [L T^{-1}] \) and the dimension of length is \( [L] \).
- Therefore, the dimension of velocity gradient is:
\[ [L T^{-1}] / [L] = [T^{-1}] \]
- Since the dimension of the velocity gradient is \( [T^{-1}] \), it is not dimensionless.

### Step 3: Analyze the second option - Stress
- Stress is defined as force per unit area.
- Mathematically, it can be expressed as:
\[ \text{Stress} = \frac{\text{Force}}{\text{Area}} \]
- The dimension of force is \( [M L T^{-2}] \) and the dimension of area is \( [L^2] \).
- Therefore, the dimension of stress is:
\[ [M L T^{-2}] / [L^2] = [M L^{-1} T^{-2}] \]
- Since the dimension of stress includes mass and length, it is not dimensionless.

### Step 4: Analyze the third option - Force Gradient
- The force gradient is defined as force per unit length.
- Mathematically, it can be expressed as:
\[ \text{Force Gradient} = \frac{\text{Force}}{\text{Length}} \]
- The dimension of force is \( [M L T^{-2}] \) and the dimension of length is \( [L] \).
- Therefore, the dimension of force gradient is:
\[ [M L T^{-2}] / [L] = [M T^{-2}] \]
- Since the dimension of force gradient includes mass, it is not dimensionless.

### Step 5: Analyze the fourth option - Angle
- The angle is defined as the ratio of arc length to radius.
- Mathematically, it can be expressed as:
\[ \text{Angle} = \frac{\text{Arc Length}}{\text{Radius}} \]
- The dimension of arc length is \( [L] \) and the dimension of radius is also \( [L] \).
- Therefore, the dimension of angle is:
\[ [L] / [L] = [1] \quad \text{(dimensionless)} \]
- Since the angle has no dimensions, it is a dimensionless quantity.

### Conclusion
The only dimensionless physical quantity among the options is the angle.

### Final Answer
**The dimensionless physical quantity is: Angle.**

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