Which of the following pairs has the same dimensions?

Options

A. Torque and work
B. Angular momentum and Planck's constant
C. Energy momentum and Planck's constant
D. Light year and wavelength

Explanation

To determine which of the given pairs has the same dimensions, we will analyze each pair one by one.

### Step 1: Analyze Torque and Work
- **Torque (τ)** is defined as the product of force (F) and the distance (r) from the pivot point. The formula is:
\[ \tau = F \cdot r \]
The dimension of force (F) is given by:
\[ F = m \cdot a = m \cdot \frac{L}{T^2} \implies [F] = M L T^{-2} \]
Therefore, the dimension of torque is:
\[ [\tau] = [F] \cdot [r] = (M L T^{-2}) \cdot L = M L^2 T^{-2} \]

- **Work (W)** is also defined as the product of force and displacement:
\[ W = F \cdot d \]
Thus, the dimension of work is:
\[ [W] = [F] \cdot [d] = (M L T^{-2}) \cdot L = M L^2 T^{-2} \]

Since both torque and work have the same dimensions:
\[ [\tau] = [W] = M L^2 T^{-2} \]

### Step 2: Analyze Angular Momentum and Planck Constant
- **Angular Momentum (L)** is defined as:
\[ L = r \cdot p = r \cdot (m \cdot v) = r \cdot (m \cdot \frac{L}{T}) \]
Thus, the dimension of angular momentum is:
\[ [L] = [r] \cdot [p] = L \cdot (M L T^{-1}) = M L^2 T^{-1} \]

- **Planck Constant (h)** has the dimension:
\[ [h] = E \cdot T = (M L^2 T^{-2}) \cdot T = M L^2 T^{-1} \]

Since both angular momentum and Planck constant have the same dimensions:
\[ [L] = [h] = M L^2 T^{-1} \]

### Step 3: Analyze Energy Momentum and Planck Constant
- **Energy (E)** has the dimension:
\[ [E] = M L^2 T^{-2} \]

- **Momentum (p)** has the dimension:
\[ [p] = m \cdot v = M \cdot \frac{L}{T} = M L T^{-1} \]

Since energy and momentum do not have the same dimensions, this pair is not valid.

### Step 4: Analyze Light Year and Wavelength
- **Light Year** is a measure of distance, hence:
\[ [\text{Light Year}] = L \]

- **Wavelength** is also a measure of distance, hence:
\[ [\text{Wavelength}] = L \]

Since both light year and wavelength have the same dimensions:
\[ [\text{Light Year}] = [\text{Wavelength}] = L \]

### Conclusion
The pairs that have the same dimensions are:
1. Torque and Work
2. Angular Momentum and Planck Constant
3. Light Year and Wavelength

### Summary of Pairs with Same Dimensions:
- Torque and Work: \( M L^2 T^{-2} \)
- Angular Momentum and Planck Constant: \( M L^2 T^{-1} \)
- Light Year and Wavelength: \( L \)