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. $\left[ \text{M}{{\text{L}}^{\text{2}}}{{\text{T}}^{\text{-1}}} \right]\text{ and }\left[ C. \text{M}{{\text{L}}^{2}}{{\text{T}}^{\text{-1}}} \right]$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. $\left[ \text{ML}{{\text{T}}^{\text{-1}}} \right]\text{ and }\left[ \text{ML}{{\text{T}}^{\text{-2}}} \right]$

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.

The energy radiated by an electromagnetic wave for example light at a particular frequency $\text{( }\!\!\nu\!\!\text{ )}$ is given by $\text{E}=\text{h }\!\!\nu\!\!\text{ }$, so the Planck constant can be expressed as the ratio of energy and frequency, $\text{h}=\text{E/ }\!\!\nu\!\!\text{ }$. 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,
$h=\dfrac{\left[ \text{M}{{\text{L}}^{\text{2}}}{{\text{T}}^{\text{-2}}} \right]}{\left[ {{\text{T}}^{\text{-1}}} \right]}=\left[ \text{M}{{\text{L}}^{\text{2}}}{{\text{T}}^{\text{-1}}} \right]$
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 $\text{ }\!\!\omega\!\!\text{ }$ is given by the formula $\text{L}=\text{I }\!\!\omega\!\!\text{ }$, 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.