Chapter 6.2, Problem 41E

a.

To determine

To find: the equation for angular displacement of the pendulum.

Answer to Problem 41E

  $\theta =\frac{\pi }{4}\cos \left(\pi t\right)$

Explanation of Solution

Given information:

The equation given for angular displacement $\left(\theta \right)$ is shown below.

  $\theta ={\theta }_m\cos \omega t$

Here ${\theta }_m,\omega ,$ and t is rotation of pendulum (in radians), angular frequency (in radians per second) and time respectively.

Angular frequency of a certain torsion pendulum $=\pi \text{ radians per second}$

Initial rotation $=\frac{\pi }{4}\text{ radians}$

Calculation:

The equation for angular displacement of the pendulum is evaluated as follows:

  $\begin{array}{c}\theta ={\theta }_m\cos \omega t\\ =\left(\frac{\pi }{4}\right)\cos \left(\pi \right)t\left[\text{Put the values of }{\theta }_m\text{ and }\omega \right]\\ =\frac{\pi }{4}\cos \left(\pi t\right)\end{array}$

Thus the equation for angular displacement of the pendulum is $\theta =\frac{\pi }{4}\cos \left(\pi t\right)$ .

b.

To determine

To find: the first two values of t.

Answer to Problem 41E

  $0.5\text{ second, and 1}\text{.5 second}$

Explanation of Solution

Given information:

The equation given for angular displacement $\left(\theta \right)$ is shown below.

  $\begin{array}{l}\theta ={\theta }_m\cos \omega t\\ \theta =\frac{\pi }{4}\cos \left(\pi t\right)\end{array}$

Here ${\theta }_m,\omega ,$ and t is rotation of pendulum (in radians), angular frequency (in radians per second) and time respectively.

Angular displacement of pendulum, $\theta =0$

Calculation:

The first two values of time $\left(t\right)$ are evaluated as follows:

  $\begin{array}{c}\theta =\frac{\pi }{4}\cos \left(\pi t\right)\\ 0=\frac{\pi }{4}\cos \left(\pi t\right)\left[\text{Put the value of }\theta \right]\\ 0=\cos \left(\pi t\right)\\ n\frac{\pi }{2}=\pi t\left[\because \cos \left(n\frac{\pi }{2}\right)=0,\text{ where }n\text{ is an odd integer}\right]\\ n=1:\to \frac{1}{2}\to 0.5\\ n=3:\to \frac{3}{2}\to 1.5\end{array}$

Thus the time is 0.5 second and 1.5 second.