College Physics by Openstax Chapter 2 Problem 15

The planetary model of the atom pictures electrons orbiting the atomic nucleus much as planets orbit the Sun. In this model, you can view hydrogen, the simplest atom, as having a single electron in a circular orbit 1.06×10-10 m in diameter.

(a) If the average speed of the electron in this orbit is known to be 2.20×106 m/s, calculate the number of revolutions per second it makes about the nucleus.

(b) What is the electron’s average velocity?

Solution:

Part A

The formula to be used is

average speed=distancetimer=dt\begin{align*} \text{average speed} & =\frac{\text{distance}}{\text{time}} \\ r & = \frac{d}{t} \end{align*}

Rearranging the formula–solving for the distance

d=r×t\begin{align*} d=r\times t \end{align*}

Substituting the given values for 1 second period

d=(2.20×106 m/s)(1 s)=2.20×106 meters\begin{align*} d & = \left(2.20\times 10^6\:\text{m/s}\right)\left(1\:\text{s}\right) \\ & =2.20\times 10^6\:\text{meters} \end{align*}

This is the total distance traveled in 1 sec.

With the given radius, the total distance traveled in 1 revolution is

1 revolution=2πr=πd=π(1.06×10−10m)\begin{align*} 1\:\text{revolution} & =2\pi \text{r} \\ & =\pi \text{d} \\ &=\pi \left(1.06\times 10^{-10}\text{m}\right) \end{align*}

Therefore, the total number of revolutions traveled in 1 second is

no. of revolutions=total distancedistance in 1 revolution=2.20×106π(1.06×10−10)=6.61×1015 revolutions  (Answer)\begin{align*} \text{no. of revolutions} & = \frac{\text{total distance}}{\text{distance in 1 revolution}} \\ & = \frac{2.20\times 10^6}{\pi \left(1.06\times 10^{-10}\right)} \\ & =6.61\times 10^{15} \ \text{revolutions} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}

Part B

In one complete revolution, the electron will go back to its original position. Thus, there is no net displacement. Therefore,

v‾=ΔxΔtv‾=0 m/s  (Answer)\begin{align*} \overline{v} & =\frac{\Delta x}{\Delta t} \\ \overline{v} & =0 \ \text{m/s} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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