The dimensions of the quantities in one (or more) of the following pairs are the same . Identify the pair(s)

Options

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

Explanation

To solve the question, we need to identify which pairs of quantities have the same dimensions. Let's analyze each pair step by step.

### Step 1: Analyze Torque and Work
- **Torque (τ)** is defined as τ = R × F, where R is the distance (radius) and F is the force.
- Dimension of R (length) = L
- Dimension of F (force) = MLT⁻²
- Therefore, the dimension of Torque:
\[ [τ] = [R] \cdot [F] = L \cdot (MLT^{-2}) = ML^2T^{-2} \]

- **Work (W)** is defined as W = F × d, where d is the displacement.
- Dimension of Work:
\[ [W] = [F] \cdot [d] = (MLT^{-2}) \cdot L = ML^2T^{-2} \]

- **Conclusion**: The dimensions of Torque and Work are the same: \( ML^2T^{-2} \).

### Step 2: Analyze Angular Momentum and Work
- **Angular Momentum (L)** is defined as L = R × P, where P is momentum (P = mv).
- Dimension of P (momentum) = MLT⁻¹
- Therefore, the dimension of Angular Momentum:
\[ [L] = [R] \cdot [P] = L \cdot (MLT^{-1}) = ML^2T^{-1} \]

- **Work** has already been calculated as:
\[ [W] = ML^2T^{-2} \]

- **Conclusion**: The dimensions of Angular Momentum and Work are not the same.

### Step 3: Analyze Energy and Young's Modulus
- **Energy (E)** is defined as E = ½mv².
- Dimension of v (velocity) = LT⁻¹
- Therefore, the dimension of Energy:
\[ [E] = M \cdot (LT^{-1})^2 = ML^2T^{-2} \]

- **Young's Modulus (Y)** is defined as stress/strain.
- Stress = Force/Area = (MLT⁻²)/(L²) = MLT⁻²L⁻² = ML⁻¹T⁻²
- Strain is dimensionless (change in length/original length).
- Therefore, the dimension of Young's Modulus:
\[ [Y] = [Stress] = ML^{-1}T^{-2} \]

- **Conclusion**: The dimensions of Energy and Young's Modulus are not the same.

### Step 4: Analyze Light Year and Wavelength
- **Light Year** is a measure of distance (the distance light travels in one year).
- Dimension of Light Year = L

- **Wavelength (λ)** is also a measure of distance.
- Dimension of Wavelength = L

- **Conclusion**: The dimensions of Light Year and Wavelength are the same: L.

### Final Conclusion
The pairs with the same dimensions are:
1. Torque and Work
2. Light Year and Wavelength