Choose the pair physical quantities which have identical dimensions

Options

A. Impulse & Linear momentum
B. Planck's constant & Angular momentum
C. M. I & Moment of force
D. Youngs modulus & Pressure

Answer

To solve the problem of identifying pairs of physical quantities that have identical dimensions, we will analyze each option provided in the question step by step.

### Step-by-Step Solution:

1. **Understanding Impulse and Linear Momentum (Option A)**:
- Impulse (J) is defined as the change in momentum (Δp).
- Linear momentum (p) is given by the formula \( p = mv \) (mass times velocity).
- The dimension of momentum is:
\[ [p] = [m][v] = M \cdot (L T^{-1}) = M L T^{-1} \]
- Since impulse is equal to momentum, their dimensions are identical:
\[ [J] = [Δp] = M L T^{-1} \]
- **Conclusion**: Option A has identical dimensions.

2. **Understanding Planck's Constant and Angular Momentum (Option B)**:
- Planck's constant (h) can be expressed as energy (E) per frequency (ν):
\[ [h] = \frac{[E]}{[ν]} = \frac{M L^2 T^{-2}}{T^{-1}} = M L^2 T^{-1} \]
- Angular momentum (L) is defined as momentum times distance:
\[ [L] = [p][r] = (M L T^{-1})(L) = M L^2 T^{-1} \]
- **Conclusion**: Both Planck's constant and angular momentum have the same dimensions \( M L^2 T^{-1} \). Thus, Option B has identical dimensions.

3. **Understanding Moment of Inertia and Moment of Force (Option C)**:
- Moment of inertia (I) is given by \( I = m r^2 \):
\[ [I] = [m][r^2] = M L^2 \]
- Moment of force (torque, τ) is given by \( τ = F \cdot r \):
\[ [τ] = [F][r] = (M L T^{-2})(L) = M L^2 T^{-2} \]
- **Conclusion**: The dimensions of moment of inertia and moment of force are different. Hence, Option C does not have identical dimensions.

4. **Understanding Young's Modulus and Pressure (Option D)**:
- Young's modulus (Y) is defined as stress (force per unit area) over strain (dimensionless):
\[ [Y] = \frac{[F]}{[A]} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]
- Pressure (P) is also defined as force per unit area:
\[ [P] = \frac{[F]}{[A]} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]
- **Conclusion**: Both Young's modulus and pressure have the same dimensions \( M L^{-1} T^{-2} \). Thus, Option D has identical dimensions.

### Final Conclusion:
The pairs of physical quantities that have identical dimensions are:
- Option A: Impulse and Linear Momentum
- Option B: Planck's Constant and Angular Momentum
- Option D: Young's Modulus and Pressure