Which of the following pairs of units are used for measuring the same physical…

Concept: Two units can be said to measure the same physical quantity only when they reduce to the identical combination of SI base units — the same dimensional formula. To compare a pair of units, express both in terms of kg, m and s and check whether the combinations match.

1. kg·m/s2 is the SI derived unit of force, the newton (N); its dimensional formula is [M L T-2].

2. N·m works out to (kg·m/s2)·m = kg·m2/s2, the unit of energy/work, the joule (J), with dimensional formula [M L2 T-2]. Since force and energy have different dimensional formulas, the pair kg·m/s2 and N·m does not match.

3. kg·m2/s2 is exactly the joule (J). Since watt W = J/s = kg·m2/s3, a watt-second Ws = W × s = kg·m2/s3 × s = kg·m2/s2 — the same combination as the joule. So kg·m2/s2 and Ws both reduce to kg·m2/s2, the unit of energy.

4. Checking the remaining pairs: kg·m/s2 (force, newton) with Ws (energy, joule) do not match, and kg·m2/s2 (energy, joule) with J/s (power, watt) do not match either, since power is energy divided by time.

Cross-check: kg·m2/s2 has dimension [M L2 T-2], and Ws = (J/s) × s = J shares that exact same dimension, [M L2 T-2] — energy. kg·m/s2 has dimension [M L T-2] (force) and J/s has dimension [M L2 T-3] (power); neither matches [M L2 T-2]. N·m also reduces to [M L2 T-2] (as derived above, in step 2), but no option pairs it with another [M L2 T-2] unit — its only partner in the option set is kg·m/s2 (force), a mismatch. The one pair that is internally consistent is kg·m2/s2 with Ws, both measuring energy (work).