The physical quantites not having same dimensions are

Options

A. momentum and planck's constant
B. speed and `(mu_0 lambda_0)^(-1//2)`
C. speed and `sqrt(p// rho)`
D. surface tension and spring constant

Explanation

To determine 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
We have the following pairs of physical quantities to analyze:
1. Momentum (P) and Planck's constant (H)
2. Speed and \( \frac{P}{\rho} \) (where P is pressure and \( \rho \) is density)
3. Surface tension (T) and spring constant (k)

### Step 2: Calculate the Dimensions of Momentum
Momentum (P) is defined as the product of mass and velocity.
- Dimension of mass (M) = [M]
- Dimension of velocity (v) = [L T\(^{-1}\)]

Thus, the dimension of momentum is:
\[ [P] = [M][L T^{-1}] = [M L T^{-1}] \]

### Step 3: Calculate the Dimensions of Planck's Constant
Planck's constant (H) can be expressed in terms of energy and frequency.
- Energy (E) has dimensions [M L\(^2\) T\(^{-2}\)]
- Frequency (ν) has dimensions [T\(^{-1}\)]

Using the relation \( E = H \nu \), we get:
\[ [H] = \frac{[E]}{[\nu]} = \frac{[M L^2 T^{-2}]}{[T^{-1}]} = [M L^2 T^{-1}] \]

### Step 4: Compare the Dimensions of Momentum and Planck's Constant
- Dimension of momentum: [M L T\(^{-1}\)]
- Dimension of Planck's constant: [M L\(^2\) T\(^{-1}\)]

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

### Step 5: Calculate the Dimensions of Speed
Speed is defined as distance traveled per unit time.
- Dimension of speed (v) = [L T\(^{-1}\)]

### Step 6: Calculate the Dimensions of \( \frac{P}{\rho} \)
Pressure (P) is defined as force per unit area.
- Dimension of force (F) = [M L T\(^{-2}\)]
- Area (A) = [L\(^2\)]

Thus, the dimension of pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{[M L T^{-2}]}{[L^2]} = [M L^{-1} T^{-2}] \]

Density (\( \rho \)) is defined as mass per unit volume.
- Volume (V) = [L\(^3\)]

Thus, the dimension of density is:
\[ [\rho] = \frac{[M]}{[L^3]} = [M L^{-3}] \]

Now, we can find the dimensions of \( \frac{P}{\rho} \):
\[ \frac{P}{\rho} = \frac{[M L^{-1} T^{-2}]}{[M L^{-3}]} = [L^2 T^{-2}] \]

### Step 7: Compare the Dimensions of Speed and \( \frac{P}{\rho} \)
- Dimension of speed: [L T\(^{-1}\)]
- Dimension of \( \frac{P}{\rho} \): [L\(^2\) T\(^{-2}\)]

Since these dimensions are also different, speed and \( \frac{P}{\rho} \) do not have the same dimensions.

### Step 8: Calculate the Dimensions of Surface Tension and Spring Constant
Surface tension (T) is defined as force per unit length:
\[ [T] = \frac{[F]}{[L]} = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]

Spring constant (k) is defined as force per unit extension:
\[ [k] = \frac{[F]}{[L]} = \frac{[M L T^{-2}]}{[L]} = [M T^{-2}] \]

### Step 9: Compare the Dimensions of Surface Tension and Spring Constant
- Dimension of surface tension: [M T\(^{-2}\)]
- Dimension of spring constant: [M T\(^{-2}\)]

Since these dimensions are the same, surface tension and spring constant do have the same dimensions.

### Final Conclusion
The physical quantities that do not have the same dimensions are:
- Momentum and Planck's constant
- Speed and \( \frac{P}{\rho} \)

Thus, the answer is that the first pair (momentum and Planck's constant) and the second pair (speed and \( \frac{P}{\rho} \)) do not have the same dimensions.