Chapter 7.5, Problem 19Q

Solution

To state: The total distance that the pendulum swings if it is released to swing freely and travels 25 inches on the first swing. On each successive swing, the pendulum travels 85 percent as far as the previous swing.

The resultant distance is $166.\overline{6}$ inches.

Given information:

A pendulum released to swing freely and travels 25 inches on the first swing. On each successive swing, the pendulum travels 85 percent as far as the previous swing.

Explanation:

On the first swing pendulum travels: $a_1=25$ inches

On the second swing pendulum travels 85 percent as far as previous swing:

  $\begin{array}{c}{a}_2=\frac{85}{100}\cdot 25\\ =0.85\cdot 25\end{array}$

Similarly, on the third swing pendulum travels 85 percent as far as previous swing:

  $a_3={\left(0.85\right)}^2\cdot 25$

It can be seen that the distance forms a geometric series with the first term 25 and common ratio $r=0.85$ .

Now the total distance is:

  $\begin{array}{c}{\displaystyle \sum _{n=0}^{\infty }25{\left(0.85\right)}^n}=\frac{25}{1-0.85}\\ =\frac{25}{0.15}\\ =166.\overline{6}\end{array}$

So, the total distance travelled by the pendulum is $166.\overline{6}$ inches.