Chapter 10.1, Problem 29PS

Solution

To find: The greatest horizontal distance travelled by a simple pendulum.

The greatest horizontal distance travelled by the pendulum is 4 inches.

Given data: The function representing horizontal distance travelled by the pendulum from the horizontal rest is modelled as, $d\left(t\right)=4\cos \pi t$ .

Method/Formula used:

A periodic function $d\left(t\right)$ representing displacement of a pendulum with time period T , frequency n , and maximum horizontal displacement (amplitude) $d_{\max .}$ (in inch) is given by,

  $\left.\begin{array}{l}d\left(t\right)=d_{\max .}\cos 2\pi nt\\ OR\\ d\left(t\right)=d_{\max .}\cos \theta \end{array}\right\}$ … (1)

Here, $\theta$ is the angle pendulum makes with horizontal as shown in Fig.1.

  

Calculations:

The function representing horizontal distance travelled by the pendulum from the horizontal rest is modelled as,

  $d\left(t\right)=4\cos \pi t$ … (2)

Comparing results (1) and (2),

  $d_{\max .}=4$

Moreover, cosine function $\cos \pi t$ has maximum value 1, {in Fig. 1 when $\theta =0$ } therefore,

  $\begin{array}{c}{d}_{\max .}=4\cdot \left(1\right)\\ =4\end{array}$

The graph of function $d\left(t\right)=4\cos \pi t$ is shown in Fig.2.

  

Thus, the greatest horizontal distance travelled by the pendulum is 4 inches.