In which of the following pairs, the two physical quantities have different dimensions?

Options

A. Planck's constant and angular momentum
B. Impulse and linear momentum
C. Moment of inertia and moment of a force
D. Energy and torque

Explanation

To determine which pair of physical quantities has different dimensions, we will analyze the dimensions of each pair provided in the options.

**Step 1: Analyze Option A - Planck's Constant and Linear Momentum**

1. **Planck's Constant (h)**:
- The relationship is given by \( E = h \nu \), where \( E \) is energy and \( \nu \) is frequency.
- The dimension of energy \( [E] \) is \( [M L^2 T^{-2}] \).
- The dimension of frequency \( [\nu] \) is \( [T^{-1}] \).
- Therefore, the dimension of Planck's constant \( [h] \) is:
\[ [h] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]

2. **Linear Momentum (p)**:
- Linear momentum is defined as \( p = mv \), where \( m \) is mass and \( v \) is velocity.
- The dimension of mass \( [m] \) is \( [M] \) and the dimension of velocity \( [v] \) is \( [L T^{-1}] \).
- Therefore, the dimension of linear momentum \( [p] \) is:
\[ [p] = [M][L T^{-1}] = [M L T^{-1}] \]

Since both have the same dimension \( [M L^2 T^{-1}] \), they are not different.

**Step 2: Analyze Option B - Impulse and Linear Momentum**

1. **Impulse (I)**:
- Impulse is defined as \( I = F \cdot t \), where \( F \) is force and \( t \) is time.
- The dimension of force \( [F] \) is \( [M L T^{-2}] \) and the dimension of time \( [t] \) is \( [T] \).
- Therefore, the dimension of impulse \( [I] \) is:
\[ [I] = [F][t] = [M L T^{-2}][T] = [M L T^{-1}] \]

2. **Linear Momentum (p)**:
- As calculated earlier, the dimension of linear momentum \( [p] \) is:
\[ [p] = [M L T^{-1}] \]

Since both have the same dimension \( [M L T^{-1}] \), they are not different.

**Step 3: Analyze Option C - Moment of Inertia and Moment of Force (Torque)**

1. **Moment of Inertia (I)**:
- Moment of inertia is defined as \( I = \frac{L}{\omega} \), where \( \omega \) is angular frequency.
- The dimension of angular momentum \( [L] \) is \( [M L^2 T^{-1}] \) and the dimension of angular frequency \( [\omega] \) is \( [T^{-1}] \).
- Therefore, the dimension of moment of inertia \( [I] \) is:
\[ [I] = \frac{[M L^2 T^{-1}]}{[T^{-1}]} = [M L^2] \]

2. **Moment of Force (Torque, τ)**:
- Torque is defined as \( \tau = r \cdot F \), where \( r \) is distance and \( F \) is force.
- The dimension of distance \( [r] \) is \( [L] \) and the dimension of force \( [F] \) is \( [M L T^{-2}] \).
- Therefore, the dimension of torque \( [\tau] \) is:
\[ [\tau] = [L][M L T^{-2}] = [M L^2 T^{-2}] \]

Since the dimensions of moment of inertia \( [M L^2] \) and moment of force (torque) \( [M L^2 T^{-2}] \) are different, this pair has different dimensions.

**Step 4: Analyze Option D - Energy and Torque**

1. **Energy (E)**:
- The dimension of energy \( [E] \) is \( [M L^2 T^{-2}] \).

2. **Torque (τ)**:
- As calculated earlier, the dimension of torque \( [\tau] \) is \( [M L^2 T^{-2}] \).

Since both have the same dimension \( [M L^2 T^{-2}] \), they are not different.

**Conclusion**: The only pair with different dimensions is **Option C**: Moment of Inertia and Moment of Force (Torque).