The physical quantities not having same dimensions are

Options

A. Pressure and energy density
B. Torque and work
C. Momentum and Planck's constant
D. Stress and Young's modulus

Answer

To determine which pair of physical quantities does not have the same dimensions, we will analyze each option step by step.

### Step 1: Analyze Pressure and Energy Density
- **Pressure (P)** is defined as force per unit area.
- Dimension of Force (F) = Mass (M) × Acceleration (A) = MLT^-2
- Area (A) = L^2
- Therefore, the dimension of Pressure = F/A = (MLT^-2) / (L^2) = ML^-1T^-2.

- **Energy Density (E_d)** is defined as energy per unit volume.
- Dimension of Energy (E) = ML^2T^-2
- Volume (V) = L^3
- Therefore, the dimension of Energy Density = E/V = (ML^2T^-2) / (L^3) = ML^-1T^-2.

**Conclusion**: Pressure and Energy Density have the same dimensions (ML^-1T^-2).

### Step 2: Analyze Torque and Work
- **Torque (τ)** is defined as the product of force and distance.
- Torque = r × F
- Dimension of Torque = L × (MLT^-2) = ML^2T^-2.

- **Work (W)** is defined as force times displacement.
- Work = F × d
- Dimension of Work = (MLT^-2) × L = ML^2T^-2.

**Conclusion**: Torque and Work have the same dimensions (ML^2T^-2).

### Step 3: Analyze Momentum and Planck Constant
- **Momentum (p)** is defined as mass times velocity.
- Dimension of Momentum = M × (LT^-1) = MLT^-1.

- **Planck Constant (h)** can be derived from the equation E = hν.
- Rearranging gives h = E/ν.
- Dimension of Planck Constant = (ML^2T^-2) / (T^-1) = ML^2T^-1.

**Conclusion**: Momentum (MLT^-1) and Planck Constant (ML^2T^-1) do not have the same dimensions.

### Step 4: Analyze Stress and Young's Modulus
- **Stress (σ)** is defined as force per unit area.
- Dimension of Stress = F/A = (MLT^-2) / (L^2) = ML^-1T^-2.

- **Young's Modulus (Y)** is defined as the ratio of stress to strain.
- Strain is dimensionless (change in length/original length).
- Therefore, the dimension of Young's Modulus = Stress = ML^-1T^-2.

**Conclusion**: Stress and Young's Modulus have the same dimensions (ML^-1T^-2).

### Final Answer
The pair of physical quantities that do not have the same dimensions is **Momentum and Planck Constant** (Option C).

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