Which of the following pairs have different dimensions?

Options

A. `Frequency and angular velocity.
B. Tension and surface tension.
C. Density and energy density.
D. Linearmomentum and angular momentum.

Explanation

To determine which pairs have different dimensions, we will analyze each pair provided in the question step by step.

### Step 1: Analyze Frequency and Angular Frequency
- **Frequency (f)** is defined as the number of cycles per second. Its dimension can be derived as follows:
\[ [f] = \frac{1}{T} = T^{-1} \]

- **Angular Frequency (ω)** is defined as the angle (in radians) per unit time. Since radians are dimensionless, we have:
\[ [\omega] = \frac{\text{angle}}{T} = \frac{1}{T} = T^{-1} \]

- **Conclusion**: Both frequency and angular frequency have the same dimensions: \( T^{-1} \).

### Step 2: Analyze Tension and Surface Tension
- **Tension (T)** is a force, and its dimension is:
\[ [T] = MLT^{-2} \]

- **Surface Tension (σ)** is defined as force per unit length:
\[ [\sigma] = \frac{F}{L} = \frac{MLT^{-2}}{L} = MT^{-2} \]

- **Conclusion**: Tension and surface tension have different dimensions: \( MLT^{-2} \) vs \( MT^{-2} \).

### Step 3: Analyze Density and Energy Density
- **Density (ρ)** is mass per unit volume:
\[ [\rho] = \frac{M}{L^3} = ML^{-3} \]

- **Energy Density (ε)** is energy per unit volume:
\[ [\epsilon] = \frac{E}{V} = \frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2} \]

- **Conclusion**: Density and energy density have different dimensions: \( ML^{-3} \) vs \( ML^{-1}T^{-2} \).

### Step 4: Analyze Linear Momentum and Angular Momentum
- **Linear Momentum (p)** is defined as mass times velocity:
\[ [p] = M \cdot (LT^{-1}) = MLT^{-1} \]

- **Angular Momentum (L)** is defined as mass times velocity times position:
\[ [L] = M \cdot (LT^{-1}) \cdot L = ML^2T^{-1} \]

- **Conclusion**: Linear momentum and angular momentum have different dimensions: \( MLT^{-1} \) vs \( ML^2T^{-1} \).

### Final Conclusion
From the analysis, the pairs that have different dimensions are:
- Tension and Surface Tension
- Density and Energy Density
- Linear Momentum and Angular Momentum

Thus, the pairs with different dimensions are:
- **Option 2**: Tension and Surface Tension
- **Option 3**: Density and Energy Density
- **Option 4**: Linear Momentum and Angular Momentum