Question

The ratio of the dimensions of Planck's constant and that of the moment of inertia is the dimension of:
A. Time
B. Frequency
C. Angular momentum
D. Velocity

Answer 1

Hint: Planck's constant is an elementary quantum of action. We can define it as a product of energy and time. So it’s unit is Js.
And moment of inertia measures the rotational inertia of the body. It’s unit is $kg{{m}^{2}}$ in SI system.By using units we can find dimensional formula and then find required ratio.

Complete step-by-step answer:
First of all we have to find out the dimensions of Planck's constant.
We know that Planck’s theory
\[E=h\mathbf{\nu }\]
Where E = energy
h = planck's constant
v = frequency
[this formula is used in the topic of light, sound etc. and so it can help us to find the dimensional formula of h]
Dimensional formula for\[E=\text{ }\!\![\!\!\text{ M}{{\text{L}}^{2}}{{\text{T}}^{-2}}\text{ }\!\!]\!\!\text{ }\] And \[\mathbf{\nu }=\dfrac{1}{T}={{\left[ T \right]}^{-1}}\]
Dimensional formula for planck's constant
\[\Rightarrow h=\dfrac{[{{M}^{1}}{{L}^{2}}{{T}^{-2}}]}{[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]}\]
\[\Rightarrow h=[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]\]................................................….. 1
Now, dimensional formula for
Moment of inertia,
$I=\left[ Mass \right]\times {{\left[ radius \right]}^{2}}$
\[\Rightarrow I=M\times {{L}^{2}}\]
\[\Rightarrow I=[M{{L}^{2}}]\]...........................................…. 2
From equation 1 and equation 2
The ratio of the dimensions of Planck constant and that of the moment of inertia is the dimension of
\[\Rightarrow \dfrac{[{{M}^{1}}{{L}^{2}}{{T}^{-1}}]}{[{{M}^{1}}{{L}^{2}}]}\]
$\Rightarrow \left[ {{T}^{-1}} \right]$
It represents dimensional formulas of frequency. Hence option B is correct.

Note: Frequency: It is a number of repetitions or occurrence of an event in a unit of time. It can be find as below formula:
$\Rightarrow f=\dfrac{1}{T}$
Hence it’s unit is ${{\sec }^{-1}}$. That’s why its dimensional formula is $\left[ {{T}^{-1}} \right]$.