Out of the following pairs, which one does not have identical dimensions?

Options

A. impulse and momentum
B. moment of inertia and moment of force
C. angular momentum and Planck's constant
D. work and torque.

Explanation

To solve the problem of identifying which pair of quantities does not have identical dimensions, we will analyze each pair one by one.

### Step 1: Analyze Impulse and Momentum
- **Impulse (I)** is defined as the change in momentum. The formula for impulse can be expressed as:
\[ I = \Delta p = m \Delta v \]
where \(m\) is mass and \(\Delta v\) is change in velocity.
- The dimensional formula for momentum (p) is:
\[ p = mv \implies [p] = [M][L][T^{-1}] \]
- Therefore, the dimensional formula for impulse is also:
\[ [I] = [M][L][T^{-1}] \]
- **Conclusion**: Impulse and momentum have the same dimensions.

### Step 2: Analyze Moment of Inertia and Moment of Force
- **Moment of Inertia (I)** is given by:
\[ I = m r^2 \]
where \(m\) is mass and \(r\) is distance.
- The dimensional formula for moment of inertia is:
\[ [I] = [M][L^2] \]
- **Moment of Force (Torque, τ)** is defined as:
\[ τ = r \times F \]
where \(F\) is force.
- The dimensional formula for force is:
\[ [F] = [M][L][T^{-2}] \]
- Therefore, the dimensional formula for torque is:
\[ [τ] = [L][M][L][T^{-2}] = [M][L^2][T^{-2}] \]
- **Conclusion**: The dimensional formula for moment of inertia is \([M][L^2]\) and for moment of force (torque) is \([M][L^2][T^{-2}]\). They are not the same.

### Step 3: Analyze Angular Momentum and Planck's Constant
- **Angular Momentum (L)** is given by:
\[ L = r \times p \]
where \(p\) is momentum.
- The dimensional formula for angular momentum is:
\[ [L] = [L][M][L][T^{-1}] = [M][L^2][T^{-1}] \]
- **Planck's Constant (h)** has dimensions:
\[ [h] = [E][T] = [M][L^2][T^{-2}][T] = [M][L^2][T^{-1}] \]
- **Conclusion**: Angular momentum and Planck's constant have the same dimensions.

### Step 4: Analyze Work and Torque
- **Work (W)** is defined as:
\[ W = F \cdot d \]
- The dimensional formula for work is:
\[ [W] = [M][L][T^{-2}][L] = [M][L^2][T^{-2}] \]
- **Torque (τ)** has already been calculated as:
\[ [τ] = [M][L^2][T^{-2}] \]
- **Conclusion**: Work and torque have the same dimensions.

### Final Conclusion
After analyzing all pairs, we find that the pair that does not have identical dimensions is:
- **Moment of Inertia and Moment of Force (Torque)**.

**Answer**: Moment of Inertia and Moment of Force.

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