Which of the following physical quantities do not have same dimensions ?

Options

A. pressure and stress
B. tension and surface tension
C. strain and angle
D. energy and work

Explanation

To solve the question of which physical quantities do not have the same dimensions, we will analyze each pair of quantities provided in the options.

### Step 1: Analyze Pressure and Stress
- **Pressure (P)** is defined as force per unit area.
\[ P = \frac{F}{A} \]
The dimension of force (F) is given by:
\[ [F] = M^1 L^1 T^{-2} \]
The dimension of area (A) is:
\[ [A] = L^2 \]
Therefore, the dimension of pressure is:
\[ [P] = \frac{M^1 L^1 T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]

- **Stress (σ)** is also defined as force per unit area.
\[ \sigma = \frac{F}{A} \]
Using the same dimensions as above, we find:
\[ [\sigma] = \frac{M^1 L^1 T^{-2}}{L^2} = M^1 L^{-1} T^{-2} \]

**Conclusion**: Pressure and stress have the same dimensions: \( M^1 L^{-1} T^{-2} \).

### Step 2: Analyze Tension and Surface Tension
- **Tension (T)** is a force, so its dimensions are:
\[ [T] = M^1 L^1 T^{-2} \]

- **Surface Tension (σ)** is defined as force per unit length.
\[ \text{Surface Tension} = \frac{F}{L} \]
Therefore, the dimension of surface tension is:
\[ [\text{Surface Tension}] = \frac{M^1 L^1 T^{-2}}{L} = M^1 L^{0} T^{-2} \]

**Conclusion**: Tension and surface tension have different dimensions: \( M^1 L^{1} T^{-2} \) and \( M^1 L^{0} T^{-2} \), respectively.

### Step 3: Analyze Strain and Angle
- **Strain** is a dimensionless quantity, defined as the ratio of change in length to original length.
\[ \text{Strain} = \frac{\Delta L}{L} \quad \Rightarrow \quad [\text{Strain}] = 1 \]

- **Angle** is also a dimensionless quantity, defined as the ratio of two lengths (arc length to radius).
\[ \text{Angle} = \frac{\text{Arc Length}}{\text{Radius}} \quad \Rightarrow \quad [\text{Angle}] = 1 \]

**Conclusion**: Strain and angle are both dimensionless and have the same dimensions.

### Step 4: Analyze Energy and Work
- **Energy (E)** has the dimension of:
\[ [E] = M^1 L^2 T^{-2} \]

- **Work (W)** is defined as force times displacement.
\[ W = F \cdot d \]
The dimension of work is:
\[ [W] = [F] \cdot [d] = (M^1 L^1 T^{-2}) \cdot (L^1) = M^1 L^2 T^{-2} \]

**Conclusion**: Energy and work have the same dimensions: \( M^1 L^2 T^{-2} \).

### Final Conclusion
From the analysis, the only pair of quantities that do not have the same dimensions is **Tension and Surface Tension**.

### Answer:
The physical quantities that do not have the same dimensions are **Tension and Surface Tension**.

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