Question

The angle that the string of a long pendulum makes with the vertical is shown as a function of time. What is the angular frequency of the pendulum? What is the amplitude of the pendulum's motion, in meters?

1.

The angle that the string of a long pendulum makes with the vertical is shown as a function of time. What is the angular frequency of the pendulum?

2.
What is the amplitude of the pendulum's motion, in meters?

Answer 1

1. Looking at the graph, it is clear that time for one complete oscillation i.e. time period is \(T=4.0\) sec

So angular frequency is given by

\(\omega=\frac{2 \pi}{T}=\frac{2 \pi}{4.0}=1.57 \mathrm{rad} / \mathrm{s}\)

2. We know,

\(\omega=\sqrt{\frac{g}{l}} \Rightarrow l=\frac{g}{\omega^{2}}\)

\(\therefore l=\frac{9.8}{1.57^{2}}=3.98 \mathrm{~m}\)

Maximum angular displacement of pendulum is

\(\theta=3^{\circ}=3^{\circ}\left(\frac{\pi}{180}\right) \mathrm{rad}=0.0523 \mathrm{rad}\)

Amplitude \(A=l . \theta=(3.98 m)(0.0523 r a d)=0.2081 m\)