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

Answer
Verified

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 $\text{E}=\text{h }\!\!\nu\!\!\text{ }$, 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.

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 $\text{E}=\text{h }\!\!\nu\!\!\text{ }$, 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.

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