The physical quantities not having same dimensions are

Options

A. torque and work
B. mometum and Plank's constant
C. Stress and Young's modulus
D. Speed and `(mu_0 in_0)^(-1//2)`

Answer

To solve the question regarding which physical quantities do not have the same dimensions, we will analyze the dimensions of the given physical quantities step by step.

### Step 1: Identify the Physical Quantities
The question mentions several physical quantities:
1. Torque
2. Work
3. Planck's constant
4. Momentum
5. Young's modulus

### Step 2: Determine the Dimensions of Each Quantity

**Torque (τ)**:
- Torque is defined as the product of force and distance from the pivot point.
- Formula: \( τ = F \cdot r \)
- Dimensions of Force (F): \( [F] = MLT^{-2} \)
- Dimensions of Distance (r): \( [r] = L \)
- Thus, the dimensions of Torque:
\[ [τ] = [F] \cdot [r] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \]

**Work (W)**:
- Work is defined as the product of force and displacement.
- Formula: \( W = F \cdot s \)
- Dimensions of Displacement (s): \( [s] = L \)
- Thus, the dimensions of Work:
\[ [W] = [F] \cdot [s] = (MLT^{-2}) \cdot (L) = ML^2T^{-2} \]

**Planck's Constant (h)**:
- Planck's constant relates energy and frequency.
- Formula: \( E = h \cdot f \)
- Dimensions of Energy (E): \( [E] = ML^2T^{-2} \)
- Dimensions of Frequency (f): \( [f] = T^{-1} \)
- Thus, the dimensions of Planck's constant:
\[ [h] = \frac{[E]}{[f]} = \frac{ML^2T^{-2}}{T^{-1}} = ML^2T^{-1} \]

**Momentum (p)**:
- Momentum is defined as the product of mass and velocity.
- Formula: \( p = m \cdot v \)
- Dimensions of Velocity (v): \( [v] = LT^{-1} \)
- Thus, the dimensions of Momentum:
\[ [p] = [m] \cdot [v] = (M) \cdot (LT^{-1}) = MLT^{-1} \]

**Young's Modulus (Y)**:
- Young's modulus is defined as stress divided by strain.
- Stress has the same dimensions as pressure: \( [Stress] = \frac{Force}{Area} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \)
- Strain is dimensionless.
- Thus, the dimensions of Young's modulus:
\[ [Y] = \frac{[Stress]}{[Strain]} = ML^{-1}T^{-2} \]

### Step 3: Compare the Dimensions
Now we compare the dimensions of the physical quantities:

1. Torque: \( [τ] = ML^2T^{-2} \)
2. Work: \( [W] = ML^2T^{-2} \)
3. Planck's Constant: \( [h] = ML^2T^{-1} \)
4. Momentum: \( [p] = MLT^{-1} \)
5. Young's Modulus: \( [Y] = ML^{-1}T^{-2} \)

### Step 4: Identify the Non-Matching Dimensions
From the comparison:
- Torque and Work have the same dimensions: \( ML^2T^{-2} \)
- Planck's Constant has dimensions \( ML^2T^{-1} \)
- Momentum has dimensions \( MLT^{-1} \)
- Young's Modulus has dimensions \( ML^{-1}T^{-2} \)

The quantities that do not have the same dimensions are:
- Planck's Constant and Momentum.

### Conclusion
The physical quantities that do not have the same dimensions are **Planck's constant and Momentum**.