Chapter 13.1, Problem 10P

To determine

To find: The kilometres does the bottom end of the pendulum travel in one day.

Answer to Problem 10P

The bottom end of the pendulum is travelled $345.6\pi \text{km}$ per day.

Explanation of Solution

Given information:

The clock pendulum swings back and forth once every two seconds. Its length is one meter and greatest angle it makes with the vertical is $12°$

Calculation:

Calculate the total angular per second displacement is four time the maximum displacement from the vertical

  $\begin{array}{c}\omega =4\cdot {12}^{\circ }\cdot \frac{\pi }{{180}^{\circ }}\\ =\frac{4\pi }{15}rad/\sec \end{array}$.

Convert from $\text{rad/sec}$ to $\text{rad/day}$ .

  $\begin{array}{c}\omega =\left(\frac{\text{4π}\text{rad}}{1\sec }\right)\left(\frac{60\sec }{1\min }\right)\left(\frac{\text{60}\text{min}}{\text{1}\text{hr}}\right)\left(\frac{\text{24}}{\text{1}\text{day}}\right)\\ =345,600\pi \text{rad/day}\end{array}$

Calculate distance per day.

  $\begin{array}{c}d=r\omega \\ d=345,600\pi \text{m}\left(\frac{1\text{km}}{1000\text{m}}\right)\\ d=345.6\text{km}\end{array}$

Therefore, the bottom end of the pendulum is travel $345.6\pi \text{km}$ per day.