The pair of quantities having same dimensions is

Options

A. Young's modulus and Energy
B. Impulse and Surface Tension
C. Angular momentum and Work
D. Work and Torque

Answer

To determine which pair of quantities has the same dimensions, we will analyze each pair provided in the question.

### Step-by-Step Solution:

1. **Identify the first pair: Young's Modulus and Energy**
- **Young's Modulus (Y)**:
- Defined as \( Y = \frac{\text{Stress}}{\text{Strain}} \)
- Stress is defined as \( \frac{\text{Force}}{\text{Area}} \) and Strain is a dimensionless quantity.
- Therefore, \( Y = \frac{\text{Force}}{\text{Area}} = \frac{F}{A} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \).
- **Energy (E)**:
- Energy is defined as \( E = \frac{1}{2}mv^2 \).
- Here, \( m \) is mass (M) and \( v^2 \) is \( (L/T)^2 = L^2T^{-2} \).
- Therefore, \( E = M \cdot L^2T^{-2} = ML^2T^{-2} \).
- **Conclusion**: The dimensions of Young's Modulus \( (ML^{-1}T^{-2}) \) and Energy \( (ML^2T^{-2}) \) are not the same.

2. **Identify the second pair: Impulse and Surface Tension**
- **Impulse (J)**:
- Defined as \( J = \Delta p = F \cdot \Delta t \).
- Force \( F = MLT^{-2} \) and time \( \Delta t = T \).
- Therefore, \( J = MLT^{-2} \cdot T = MLT^{-1} \).
- **Surface Tension (σ)**:
- Defined as \( \sigma = \frac{F}{L} \).
- Force \( F = MLT^{-2} \) and length \( L \).
- Therefore, \( \sigma = \frac{MLT^{-2}}{L} = MT^{-2} \).
- **Conclusion**: The dimensions of Impulse \( (MLT^{-1}) \) and Surface Tension \( (MT^{-2}) \) are not the same.

3. **Identify the third pair: Angular Momentum and Work**
- **Angular Momentum (L)**:
- Defined as \( L = mvr \).
- Here, \( m = M \), \( v = L/T \), and \( r = L \).
- Therefore, \( L = M \cdot \frac{L}{T} \cdot L = ML^2T^{-1} \).
- **Work (W)**:
- Defined as \( W = F \cdot d \).
- Force \( F = MLT^{-2} \) and distance \( d = L \).
- Therefore, \( W = MLT^{-2} \cdot L = ML^2T^{-2} \).
- **Conclusion**: The dimensions of Angular Momentum \( (ML^2T^{-1}) \) and Work \( (ML^2T^{-2}) \) are not the same.

4. **Identify the fourth pair: Work and Torque**
- **Work (W)**:
- As calculated before, \( W = ML^2T^{-2} \).
- **Torque (τ)**:
- Defined as \( τ = F \cdot r \).
- Force \( F = MLT^{-2} \) and radius \( r = L \).
- Therefore, \( τ = MLT^{-2} \cdot L = ML^2T^{-2} \).
- **Conclusion**: The dimensions of Work \( (ML^2T^{-2}) \) and Torque \( (ML^2T^{-2}) \) are the same.

### Final Answer:
The pair of quantities having the same dimensions is **Work and Torque**.

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