Out of the following pairs, which one does not have identical dimensions

To determine which pair of physical quantities does not have identical dimensions, we need to analyze the dimensional formula for each quantity in the given options.

Let the fundamental dimensions be represented by:
Mass = [M]
Length = [L]
Time = [T]

1. Moment of inertia and moment of a force:
- Moment of Inertia (I): It is given by the formula:

$I\; =\; m\; r2$

where $m$ is mass and $r$ is distance.
Therefore, the dimensional formula of Moment of Inertia is:

$[I]\; =\; [M\; L2T0]$

- Moment of a force (Torque, $\tau$): It is given by the formula:

$\tau \; =\; r\; \times \; F$

where $r$ is distance and $F$ is force ($F\; =\; mass\; \times \; acceleration$).
The dimensions of Force ($F$) are $[M\; L\; T-2]$.
Therefore, the dimensional formula of torque/moment of a force is:

$[\tau ]\; =\; [L]\; \times \; [M\; L\; T-2]\; =\; [M\; L2T-2]$

Since $[M\; L2T0]\; \ne \; [M\; L2T-2]$, this pair does not have identical dimensions.

Let us verify the other options to confirm:
2. Work and Torque:
- Work (W): $W\; =\; Force\; \times \; displacement\; =\; [M\; L\; T-2]\; \times \; [L]\; =\; [M\; L2T-2]$.
- Torque ($\tau$): $[\tau ]\; =\; [M\; L2T-2]$ (as calculated above).
Both have identical dimensions.

3. Angular momentum and Planck's constant:
- Angular momentum (L): $L\; =\; m\; v\; r\; =\; [M]\; \times \; [L\; T-1]\; \times \; [L]\; =\; [M\; L2T-1]$.
- Planck's constant (h): From $E\; =\; h\; \nu$, where $E$ is energy ($[M\; L2T-2]$) and $\nu$ is frequency ($[T-1]$):

$[h]\; =\; [E]\; /\; [\nu ]\; =\; [M\; L2T-2]\; /\; [T-1]\; =\; [M\; L2T-1]$

Both have identical dimensions.

4. Impulse and Momentum:
- Impulse (I): $I\; =\; Force\; \times \; time\; =\; [M\; L\; T-2]\; \times \; [T]\; =\; [M\; L\; T-1]$.
- Momentum (p): $p\; =\; mass\; \times \; velocity\; =\; [M]\; \times \; [L\; T-1]\; =\; [M\; L\; T-1]$.
Both have identical dimensions.

Therefore, the pair that does not have identical dimensions is Moment of inertia and moment of a force.