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 moment and work
C. energy and Young's modulus
D. light year and wavelength

Answer

To solve the problem of identifying which pairs of quantities have the same dimensions, we will analyze each pair step by step.

### Step-by-Step Solution:

1. **Pair 1: Torque and Work**
- **Torque (τ)**: Torque is defined as the product of the radius (r) and force (F). The formula is:
\[ τ = r \times F \]
- Dimension of radius (r) = \(L\)
- Dimension of force (F) = \(MLT^{-2}\)
- Therefore, the dimension of torque:
\[ [τ] = [r] \cdot [F] = L \cdot (MLT^{-2}) = ML^2T^{-2} \]

- **Work (W)**: Work is defined as the dot product of force and displacement (d). The formula is:
\[ W = F \cdot d \]
- Dimension of displacement (d) = \(L\)
- Therefore, the dimension of work:
\[ [W] = [F] \cdot [d] = (MLT^{-2}) \cdot L = ML^2T^{-2} \]
- **Conclusion**: Both torque and work have the same dimension \(ML^2T^{-2}\).

2. **Pair 2: Angular Momentum and Work**
- **Angular Momentum (L)**: Angular momentum is defined as the product of radius and linear momentum (p). The formula is:
\[ L = r \times p \]
- Dimension of linear momentum (p) = \(MLT^{-1}\)
- Therefore, the dimension of angular momentum:
\[ [L] = [r] \cdot [p] = L \cdot (MLT^{-1}) = ML^2T^{-1} \]

- **Work (W)**: As calculated above, the dimension of work is \(ML^2T^{-2}\).
- **Conclusion**: Angular momentum and work have different dimensions \(ML^2T^{-1}\) and \(ML^2T^{-2}\) respectively.

3. **Pair 3: Energy and Young's Modulus**
- **Energy (E)**: Energy can be expressed as:
\[ E = \frac{1}{2}mv^2 \]
- Dimension of mass (m) = \(M\)
- Dimension of velocity (v) = \(LT^{-1}\)
- Therefore, the dimension of energy:
\[ [E] = M \cdot (LT^{-1})^2 = ML^2T^{-2} \]

- **Young's Modulus (Y)**: Young's modulus is defined as stress divided by strain:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
- Dimension of stress = \(ML^{-1}T^{-2}\)
- Therefore, the dimension of Young's modulus:
\[ [Y] = \frac{ML^{-1}T^{-2}}{1} = ML^{-1}T^{-2} \]
- **Conclusion**: Energy and Young's modulus have different dimensions \(ML^2T^{-2}\) and \(ML^{-1}T^{-2}\) respectively.

4. **Pair 4: Light Year and Wavelength**
- **Light Year**: A light year is the distance light travels in one year. Therefore, its dimension is:
\[ [\text{Light Year}] = L \]

- **Wavelength (λ)**: Wavelength is defined as the distance between successive crests of a wave. Therefore, its dimension is also:
\[ [\text{Wavelength}] = L \]
- **Conclusion**: Both light year and wavelength have the same dimension \(L\).

### Final Conclusion:
The pairs that have the same dimensions are:
- Pair 1: Torque and Work
- Pair 4: Light Year and Wavelength

### Answer:
The correct pairs are 1 and 4.

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