Question

The ratio of the dimensions of Plancks constant and that of the moment of inertia is the dimension of

• A. frequency
• B. velocity
• C. angular momentum
• D. time

Answer 1

Answer: (A)

frequency

Answer 2

$E=h v$
$\Rightarrow h=$ Plancks constant $=\frac{E}{V}$
$\therefore[h]=\frac{[E]}{[v]}=\frac{\left[M L^{2} T^{-2}\right]}{\left[T^{-1}\right]}$
$=\left[M L^{2} T^{-1}\right]$
and $I=$ moment of inertia $=M R^{2}$
$\Rightarrow[I]=[M]\left[L^{2}\right]=\left[M L^{2}\right]$
Hence, $\frac{[h]}{[I]}=\frac{\left[M L^{2} T^{-1}\right]}{\left[M L^{2}\right]}=\left[T^{-1}\right]$
$=\frac{1}{[T]}=$ dimensions of frequency
Alternative: $\frac{h}{I}=\frac{E / v}{I}$
$=\frac{E \times T}{I}=\frac{\left(k g-m^{2} / s^{2}\right) \times s}{\left(k g-m^{2}\right)}$
$=\frac{1}{s}=\frac{1}{\text { time }}=$ frequency
Thus, dimensions of $\frac{h}{I}$ is same as of frequency.