A pair of physical quantities having same dimensional formula is

Options

A. force and torque
B. work and energy
C. force and impulse
D. linear momentum and angular momentum

Explanation

To solve the question of finding a pair of physical quantities that have the same dimensional formula, we will analyze each option provided in the question step by step.

### Step 1: Analyze Option A - Force and Torque
1. **Force (F)** is defined as \( F = ma \), where \( m \) is mass and \( a \) is acceleration.
- Dimension of mass \( [m] = M \)
- Dimension of acceleration \( [a] = L T^{-2} \)
- Therefore, the dimension of force is:
\[ [F] = M L T^{-2} \]

2. **Torque (τ)** is defined as \( τ = Iα \), where \( I \) is moment of inertia and \( α \) is angular acceleration.
- Dimension of moment of inertia \( [I] = M L^2 \)
- Dimension of angular acceleration \( [α] = T^{-2} \)
- Therefore, the dimension of torque is:
\[ [τ] = M L^2 T^{-2} \]

3. **Comparison**:
- Dimension of force: \( M L T^{-2} \)
- Dimension of torque: \( M L^2 T^{-2} \)
- **Conclusion**: They are different.

### Step 2: Analyze Option B - Work and Energy
1. **Work (W)** is defined as \( W = F \cdot d \), where \( d \) is distance.
- Dimension of distance \( [d] = L \)
- Therefore, the dimension of work is:
\[ [W] = [F] \cdot [d] = (M L T^{-2}) \cdot L = M L^2 T^{-2} \]

2. **Energy (E)** can be expressed as \( E = \frac{1}{2} mv^2 \).
- Dimension of velocity \( [v] = L T^{-1} \)
- Therefore, the dimension of energy is:
\[ [E] = M \cdot (L T^{-1})^2 = M L^2 T^{-2} \]

3. **Comparison**:
- Dimension of work: \( M L^2 T^{-2} \)
- Dimension of energy: \( M L^2 T^{-2} \)
- **Conclusion**: They are the same.

### Step 3: Analyze Option C - Force and Impulse
1. **Impulse (I)** is defined as \( I = F \cdot t \), where \( t \) is time.
- Therefore, the dimension of impulse is:
\[ [I] = [F] \cdot [t] = (M L T^{-2}) \cdot T = M L T^{-1} \]

2. **Comparison**:
- Dimension of force: \( M L T^{-2} \)
- Dimension of impulse: \( M L T^{-1} \)
- **Conclusion**: They are different.

### Step 4: Analyze Option D - Linear Momentum and Angular Momentum
1. **Linear Momentum (P)** is defined as \( P = mv \).
- Therefore, the dimension of linear momentum is:
\[ [P] = M \cdot (L T^{-1}) = M L T^{-1} \]

2. **Angular Momentum (L)** is defined as \( L = Iω \), where \( ω \) is angular velocity.
- Therefore, the dimension of angular momentum is:
\[ [L] = [I] \cdot [ω] = (M L^2) \cdot T^{-1} = M L^2 T^{-1} \]

3. **Comparison**:
- Dimension of linear momentum: \( M L T^{-1} \)
- Dimension of angular momentum: \( M L^2 T^{-1} \)
- **Conclusion**: They are different.

### Final Conclusion
After analyzing all options, we find that the only pair of physical quantities that have the same dimensional formula is **Work and Energy**.

### Answer
**B: Work and Energy**