The physical quantities not having same dimensions are

Options

A. (a) torque and Work
B. momentum and planck's constant
C. ( c ) stress and young's modulus
D. (d) speed and `(mu-(0)epsilon_(0))^(-1//2)`

Explanation

To determine which physical quantities do not have the same dimensions, we can analyze the dimensions of the given quantities step by step.

### Step 1: Identify the Quantities
We are given two pairs of physical quantities:
1. Torque and Work
2. Momentum and Planck's constant

### Step 2: Find the Dimensions of Torque
Torque (τ) is defined as the cross product of the radius (R) and force (F):
\[ \tau = R \times F \]
The dimension of force (F) is given by:
\[ F = m \cdot a = M \cdot L \cdot T^{-2} \]
where:
- M = mass
- L = length
- T = time

The dimension of torque can be expressed as:
\[ [\tau] = [R] \cdot [F] = L \cdot (M \cdot L \cdot T^{-2}) = M \cdot L^2 \cdot T^{-2} \]

### Step 3: Find the Dimensions of Work
Work (W) is defined as the dot product of force and displacement:
\[ W = F \cdot d \]
Using the same dimension for force (F) as above:
\[ [W] = [F] \cdot [d] = (M \cdot L \cdot T^{-2}) \cdot L = M \cdot L^2 \cdot T^{-2} \]

### Step 4: Compare Dimensions of Torque and Work
From the calculations:
- Dimension of Torque: \( M \cdot L^2 \cdot T^{-2} \)
- Dimension of Work: \( M \cdot L^2 \cdot T^{-2} \)

Since both quantities have the same dimensions, they are not the correct answer.

### Step 5: Find the Dimensions of Momentum
Momentum (p) is defined as:
\[ p = m \cdot v \]
where velocity (v) has the dimension:
\[ [v] = L \cdot T^{-1} \]
Thus, the dimension of momentum is:
\[ [p] = [m] \cdot [v] = M \cdot (L \cdot T^{-1}) = M \cdot L \cdot T^{-1} \]

### Step 6: Find the Dimensions of Planck's Constant
Planck's constant (h) can be derived from the equation:
\[ E = h \cdot \nu \]
where:
- E is energy
- ν is frequency

The dimension of frequency (ν) is:
\[ [\nu] = T^{-1} \]
The dimension of energy (E) can be expressed as:
\[ [E] = [F] \cdot [d] = (M \cdot L \cdot T^{-2}) \cdot L = M \cdot L^2 \cdot T^{-2} \]

Now, substituting into the equation for Planck's constant:
\[ [h] = \frac{[E]}{[\nu]} = \frac{M \cdot L^2 \cdot T^{-2}}{T^{-1}} = M \cdot L^2 \cdot T^{-1} \]

### Step 7: Compare Dimensions of Momentum and Planck's Constant
From the calculations:
- Dimension of Momentum: \( M \cdot L \cdot T^{-1} \)
- Dimension of Planck's Constant: \( M \cdot L^2 \cdot T^{-1} \)

Since the dimensions are different, we conclude that momentum and Planck's constant do not have the same dimensions.

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