Which one of the following physical quantities has the same dimensions

Which one of the following physical quantities has the same dimensions as that of Planck’s constant h?

Linear momentum
Angular momentum
Force
Potential energy
This question was previously asked in
UPSC CDS-2 – 2016
The physical quantity that has the same dimensions as Planck’s constant (h) is angular momentum.
– Planck’s constant (h) relates the energy (E) of a photon to its frequency (ν) by the equation E = hν. The dimensions of E are [ML²T⁻²] and the dimensions of ν are [T⁻¹]. Therefore, the dimensions of h are [E/ν] = [ML²T⁻²] / [T⁻¹] = [ML²T⁻¹].
– Let’s check the dimensions of the given options:
– Linear momentum (p = mv): [M] * [LT⁻¹] = [MLT⁻¹].
– Angular momentum (L = Iω or L = rp): [Moment of inertia * Angular velocity] or [Radius * Linear momentum]. Dimensions are [ML²] * [T⁻¹] or [L] * [MLT⁻¹] = [ML²T⁻¹].
– Force (F = ma): [M] * [LT⁻²] = [MLT⁻²].
– Potential energy (e.g., mgh): [M] * [LT⁻²] * [L] = [ML²T⁻²].
– Comparing the dimensions, angular momentum [ML²T⁻¹] has the same dimensions as Planck’s constant [ML²T⁻¹].
In quantum mechanics, angular momentum is quantized in units of Planck’s reduced constant (ħ = h/2π), which has the same dimensions as h. This dimensional equivalence is not a coincidence but reflects a fundamental connection between energy, frequency, and angular momentum in quantum physics.