Question

Show thath and angular momentum have the same units.

Short answer

The unit for angular momentum is$\left(J\right)$ . Since$N$ is a dimensionless quantity, both angular momentum and$h$ have same units. Thus, the units of$h$ is $J·s$.

Step-by-step solution

Definition of Angular Momentum

Angular momentum, property characterizing the rotary inertia of an object or system of objects in motion about an axis that may or may not pass through the object or system.

Show that h and angular momentum have the same units.

Bohr postulated that the angular momentum of electrons in a hydrogen atom is quantized. The translation angular momentum of the electron is an integral multiple of$h$.

Mathematically angular momentum can be expressed as

$\overrightarrow{L}=Nh.....\left(1\right)$

Here$h$ is$\frac{1}{2\pi }$ times the Planck constant$h$and $N$is the integer.

The angular momentum of a particle of mass$m$moving in circular orbit of radius$r$ with velocity$v$is given by

$\overrightarrow{L}=mvr.....\left(2\right)$

Here mass is measured in kg, velocity is measured in$m/s$and radius is measured in meters.

Explanation

The units for angular momentum is

$$\begin{gathered} \overrightarrow{L}=\left(kg\right)\left(m/s\right)\left(m\right) \\ =\left(kg·m^2/s\right)\left(\frac{s}{s}\right) \\ =\left(kg·m^2/s^2\right)·s \end{gathered}$$

As is equivalent to Joules$\left(J\right)$, the unit for angular momentum is $\left(J\right)$. Since $N$is a dimensionless quantity, both angular momentum and $h$have same units. Thus, the units of$h$ is $J·s$.