Question

The dimensional formula for Planck’s constant and angular momentum are,
A. $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-2}}\right]\text{ and }\left[\text{ML}{\text{T}}^{\text{-1}}\right]$
B. $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]\text{ and }[C.\text{M}{\text{L}}^2{\text{T}}^{\text{-1}}]$
C. $\left[\text{M}{\text{L}}^3{\text{T}}^1\right]\text{ and }\left[\text{M}{\text{L}}^2{\text{T}}^{\text{-2}}\right]$
D. $$\left[\text{ML}{\text{T}}^{\text{-1}}\right]\text{ and }\left[\text{ML}{\text{T}}^{\text{-2}}\right]$$

Answer 1

Hint: Think about how the Planck’s constant is related to energy and frequency and how angular momentum is related to moment of inertia and angular velocity.

Complete step by step answer:
The energy radiated by an electromagnetic wave for example light at a particular frequency $\text{( }\nu \text{ )}$ is given by $\text{E}=\text{h }\nu$, so the Planck constant can be expressed as the ratio of energy and frequency, $\text{h}=\text{E/ }\nu$. The dimensional formula associated with energy is $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-2}}\right]$ and the dimensional formula for frequency is $\left[{\text{T}}^{\text{-1}}\right]$.
So the dimensional formula for Planck’s constant can be derived from these,
$h={\displaystyle \frac{\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-2}}\right]}{\left[{\text{T}}^{\text{-1}}\right]}}=\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$
So the dimensional formula for Planck’s constant is $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$.
The angular momentum of a body whose moment of inertia is I and angular velocity $\omega$ is given by the formula $\text{L}=\text{I }\omega$, The dimensional formula for moment of inertia is given by $\left[\text{M}{\text{L}}^{\text{2}}\right]$ and the dimensional formula for angular velocity is $\left[{\text{T}}^{\text{-1}}\right]$ so the dimensional formula for angular momentum is the product of these two dimensional formulas,
$\text{L}=\left[\text{M}{\text{L}}^{\text{2}}\right]\left[{\text{T}}^{\text{-1}}\right]=\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$
So the dimensional formula for angular momentum is $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$.
So considering the dimensional formulas we got for Planck’s constant and angular momentum, the answer to our question will be option (B)- $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$ and $\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-1}}\right]$.
As you can see that the dimensional formula for both the quantities are same.

Note: The angular momentum can also be expressed as the product of a body of mass m, its linear velocity and the distance r from the axis. $\text{L}=\text{mvr}$.
$\left[\text{M}{\text{L}}^{\text{2}}{\text{T}}^{\text{-2}}\right]$ is the dimensional formula for Energy.
$\left[\text{ML}{\text{T}}^{\text{-2}}\right]$ is the dimensional formula for force.
$\left[\text{ML}{\text{T}}^{\text{-1}}\right]$ is the dimensional formula for momentum.