Question

The ratio of dimensions of planck constant and that of the moment of inertia has the dimension of
A). Time
B). Frequency
C). Angular momentum
D). Velocity

Answer 1

Hint: In this question we find the dimensional formula of planck's constant by using the equation of the energy of photons that is $E = h\nu $ as $\left[ {M{L^2}{T^{ - 1}}} \right]$ . Then we find the dimensional formula of moment of inertia using the formula of \[I = \sum\limits_{i = 1}^n {{M_i}x_i^2} \] as $\left[ {{M^1}{L^2}{T^0}} \right]$ . Now we find the ratio of both and get the dimensional formula of result.

Complete step-by-step solution -
First, we find the dimensional formula of planck's constant. For this, we use the equation of the energy of photons that is
$E = h\nu $
Here E is the energy of the photons whose unit is given as $\left[ {M{L^2}{T^{ - 2}}} \right]$ .
And $\nu $ is the frequency whose unit is $\left[ {{T^{ - 1}}} \right]$ .
Now we can find the planck's constant as
$h = \dfrac{E}{\nu } = \dfrac{{\left[ {M{L^2}{T^{ - 2}}} \right]}}{{\left[ {{T^{ - 1}}} \right]}}$
$h = \left[ {M{L^2}{T^{ - 1}}} \right]$
Now we know the expression of moment of inertia that is
\[I = \sum\limits_{i = 1}^n {{M_i}x_i^2} \]
Here \[{M_i}\] is the mass of the \[{i^{th}}\] particle.
 And \[{x_i}\] is the perpendicular distance of the particle from the axis of rotation.
As can be seen from the above expression that the SI unit of moment of inertia is $Kg{m^2}$
We can calculate the dimensional formula from the SI unit that is
$ \Rightarrow Kg{m^2} = \left[ {{M^1}{L^0}{T^0}} \right]\left[ {{M^0}{L^2}{T^0}} \right]$
$ \Rightarrow \left[ {{M^1}{L^2}{T^0}} \right]$
So dimensional formula of moment of inertia is $\left[ {{M^1}{L^2}{T^0}} \right]$
Now the dimensional formula of the ratio of planck's constants and moment of inertia is given as
$\dfrac{h}{I} = \dfrac{{\left[ {M{L^2}{T^{ - 1}}} \right]}}{{\left[ {{M^1}{L^2}{T^0}} \right]}} = \left[ {{T^{ - 1}}} \right]$
We know that the unit of frequency is Hertz which is second inverse that is
$f = \dfrac{1}{t} = \left[ {T{}^{ - 1}} \right]$
Hence, the dimensional formula of ratio of planck's constants and moment of inertia is the same as that of the frequency.

Note: For these types of questions we need to know the dimensional formulas of some basic constants like the moment of inertia, planck's constant, frequency, energy, force, etc. For these types of questions we first convert the formula in the form of the fundamental dimension formula.