Chapter 5.6, Problem 38E

Cock Pendulum The pendulum in a grandfather clock makes one complete swing every 2 s. The maximum angle that the pendulum makes with respect to its rest position is 10°. We know from physical principles that the angle θ between the pendulum and its rest position changes in simple harmonic fashion. Find an equation that describes the size of the angle θ as a function of lime. (Take t = 0 to be a time when the pendulum is vertical.)

To determine

To evaluate: The equation for the size angle $\theta$ in a pendulum as a function of time.

Answer to Problem 38E

The equation for the size angle in a pendulum is $f\left(t\right)=10\sin \pi t$.

Explanation of Solution

Given:

In every 2s the pendulum completes one complete swing and the maximum angle of the pendulum is $10°$ from its rest position.

Calculation:

Assume that the motion of pendulum is in simple harmonic motion.

So, the pendulum must follow the general equation for the simple harmonic motion,

$y=a\sin \left(\omega \left(t-c\right)\right)+b$ (1)

The motion of pendulum is in simple harmonic motion so, the amplitude of the pendulum is 10 and the period for the revolution is 2-s.

Calculate the value of $\omega$ by the formula,

$\omega =\frac{2\pi }{p}$

Substitute 12 for the period p in the above formula.

$\begin{array}{c}\omega =\frac{2\pi }{2}\\ =\pi \end{array}$

The value of amplitude a is $10$ and $\omega$ is $\pi$.

Substitute $10$ for a and $\pi$ for $\omega$ in equation (1),

$y=10\sin \pi t$

Thus, the equation for the size angle of the pendulum is $y=10\sin \pi t$.