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?

[amp_mcq option1=”Linear momentum” option2=”Angular momentum” option3=”Force” option4=”Potential energy” correct=”option2″]

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.