The physical quantities not having same dimensions are

The correct option is Momentum and Planck's constant.

To determine which pair of physical quantities does not share the same dimensions, we can analyze the dimensional formula for each option step-by-step.

1. Speed and $({\mu }_0{\epsilon }_0{)}^{-0.5}$

From electromagnetic theory, the speed of light in vacuum $c$ is related to the permeability of free space $\left({\mu }_0\right)$ and the permittivity of free space $\left({\epsilon }_0\right)$ by the relation:

$c=\frac{1}{\sqrt{{\mu }_0{\epsilon }_0}}=({\mu }_0{\epsilon }_0{)}^{-0.5}$

Since both quantities represent speed, their dimensional formula is:

$\left[{\mathrm{LT}}^{-1}\right]$

Therefore, this pair has the same dimensions.

2. Torque and Work

Torque $\left(\tau \right)$ is defined as the cross product of position vector and force:

$\tau =r\times F$

Its dimensional formula is:

$\left[\text{Torque}\right]=\left[L\right]\times \left[{\mathrm{MLT}}^{-2}\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$

Work $\left(W\right)$ is the dot product of force and displacement:

$W=F·d$

Its dimensional formula is:

$\left[\text{Work}\right]=\left[{\mathrm{MLT}}^{-2}\right]\times \left[L\right]=\left[{\mathrm{ML}}^2{T}^{-2}\right]$

Both torque and work have the same dimensions.

3. Momentum and Planck's constant

Linear momentum $\left(p\right)$ is the product of mass and velocity:

$p=mv$

Its dimensional formula is:

$\left[\text{Momentum}\right]=\left[M\right]\times \left[{\mathrm{LT}}^{-1}\right]=\left[{\mathrm{MLT}}^{-1}\right]$

Planck's constant $\left(h\right)$ can be derived from the energy of a photon relation $E=h\nu$, where $\nu$ is frequency:

$h=\frac{E}{\nu }$

Its dimensional formula is:

$\left[h\right]=\frac{\left[{\mathrm{ML}}^2{T}^{-2}\right]}{\left[T^{-1}\right]}$

Simplifying the division gives:

$\left[h\right]=\left[{\mathrm{ML}}^2{T}^{-1}\right]$

Comparing the two, we see that $\left[{\mathrm{MLT}}^{-1}\right]\ne \left[{\mathrm{ML}}^2{T}^{-1}\right]$. Thus, momentum and Planck's constant do not have the same dimensions.

4. Stress and Young's modulus

Stress is defined as restoring force per unit area:

$\text{Stress}=\frac{F}{A}$

Its dimensional formula is:

$\left[\text{Stress}\right]=\frac{\left[{\mathrm{MLT}}^{-2}\right]}{\left[L^2\right]}=\left[{\mathrm{ML}}^{-1}{T}^{-2}\right]$

Young's modulus $\left(Y\right)$ is the ratio of tensile stress to tensile strain:

$Y=\frac{\text{Stress}}{\text{Strain}}$

Since strain is a dimensionless quantity (ratio of change in length to original length), Young's modulus shares the exact same dimensions as stress:

$\left[Y\right]=\left[{\mathrm{ML}}^{-1}{T}^{-2}\right]$

Therefore, stress and Young's modulus have the same dimensions.