The pair of physical quantities having the same dimensions is

Options

A. Power and Torque
B. Thermal capacity and specific heat
C. Latent heat and gravitational potential
D. Angular momentum and Impulse

Answer

To solve the question of which pair of physical quantities has the same dimensions, we will follow these steps:

### Step 1: Identify the Physical Quantities
We need to identify the pairs of physical quantities given in the question. The pairs mentioned are:
1. Power and Torque
2. Thermal Capacity and Specific Heat
3. Latent Heat and Gravitational Potential
4. Angular Momentum and Impulse

### Step 2: Write Down the Dimensional Formulas
Next, we will write down the dimensional formulas for each of the physical quantities in the pairs.

1. **Power (P)**:
- Dimensional formula: \( [P] = M L^2 T^{-3} \)

2. **Torque (τ)**:
- Dimensional formula: \( [τ] = M L^2 T^{-2} \)

3. **Thermal Capacity (C)**:
- Dimensional formula: \( [C] = M^0 L^2 T^{-2} K^{-1} \)

4. **Specific Heat (c)**:
- Dimensional formula: \( [c] = M^0 L^2 T^{-2} K^{-1} \)

5. **Latent Heat (L)**:
- Dimensional formula: \( [L] = M^0 L^2 T^{-2} \)

6. **Gravitational Potential (V)**:
- Dimensional formula: \( [V] = M^0 L^2 T^{-2} \)

7. **Angular Momentum (L)**:
- Dimensional formula: \( [L] = M^1 L^2 T^{-1} \)

8. **Impulse (I)**:
- Dimensional formula: \( [I] = M^1 L^1 T^{-1} \)

### Step 3: Compare the Dimensional Formulas
Now we will compare the dimensional formulas of the pairs:

1. **Power and Torque**:
- \( [P] = M L^2 T^{-3} \)
- \( [τ] = M L^2 T^{-2} \)
- **Not the same.**

2. **Thermal Capacity and Specific Heat**:
- \( [C] = M^0 L^2 T^{-2} K^{-1} \)
- \( [c] = M^0 L^2 T^{-2} K^{-1} \)
- **Same dimensions.**

3. **Latent Heat and Gravitational Potential**:
- \( [L] = M^0 L^2 T^{-2} \)
- \( [V] = M^0 L^2 T^{-2} \)
- **Same dimensions.**

4. **Angular Momentum and Impulse**:
- \( [L] = M^1 L^2 T^{-1} \)
- \( [I] = M^1 L^1 T^{-1} \)
- **Not the same.**

### Step 4: Conclusion
From the comparisons, we find that:
- The pairs **Thermal Capacity and Specific Heat** and **Latent Heat and Gravitational Potential** both have the same dimensions.

### Final Answer
The pairs of physical quantities having the same dimensions are:
1. Thermal Capacity and Specific Heat
2. Latent Heat and Gravitational Potential

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