Vibrations and Waves

French A.P

Chapter 2

The superposition of periodic motions - all with Video Answers

Educators

Chapter Questions

Problem 1

Express the following in the form $z=\operatorname{Re}\left[A e^{i(\omega t+a)}\right]$
(a) $z=\sin \omega t+\cos \omega t$.
(b) $z=\cos (\omega t-\pi / 3)-\cos \omega t$
(c) $z=2 \sin \omega t+3 \cos \omega t$
(d) $z=\sin \omega t-2 \cos (\omega t-\pi / 4)+\cos \omega t$

Ajay Singhal

Ajay Singhal

Numerade Educator

Problem 2

A particle is simultaneously subjected to three simple harmonic motions, all of the same frequency and in the $x$ direction. If the amplitudes are $0.25,0.20$, and $0.15 \mathrm{~mm}$, respectively, and the phase difference between the first and second is $45^{\circ}$, and between the second and third is $30^{\circ}$, find the amplitude of the resultant displacement and its phase relative to the first (0.25-mm amplitude) component.

Ajay Singhal

Numerade Educator

Problem 3

Two vibrations along the same line are described by the equations
$$
\begin{aligned}
&y_{1}=A \cos 10 \pi t \\
&y_{2}=A \cos 12 \pi t
\end{aligned}
$$
Find the beat period, and draw a careful sketch of the resultant disturbance over one beat period.

Ajay Singhal

Numerade Educator

Problem 4

Find the frequency of the combined motion of each of the following:
(a) $\sin (2 \pi t-\sqrt{2})+\cos (2 \pi t)$
(b) $\sin (12 \pi t)+\cos (13 \pi t-\pi / 4)$
(c) $\sin (3 t)-\cos (\pi t)$.

Ajay Singhal

Numerade Educator

Problem 5

Two vibrations at right angles to one another are described by the equations
$$
\begin{aligned}
&x=10 \cos (5 \pi t) \\
&y=10 \cos (10 \pi t+\pi / 3)
\end{aligned}
$$
Construct the Lissajous figure of the combined motion.

Ajay Singhal

Numerade Educator

Problem 6

Construct the Lissajous figures for the following motions:
(a) $x=\cos 2 \omega t, y=\sin 2 \omega t$
(b) $x=\cos 2 \omega t, y=\cos (2 \omega t-\pi / 4)$.
(c) $x=\cos 2 \omega t, y=\cos \omega t$

Ajay Singhal

Numerade Educator