O Marie Cornu, a physicist at the Polytechnic Institute in...

O Marie Cornu, a physicist at the Polytechnic Institute in Paris, invented phasors in about 1880 . This problem helps you see their general utility in representing oscillations. Two mechanical vibrations are represented by the expressions $$ y_{1}=(12.0 \mathrm{cm}) \sin (4.5 t) $$ and $$ y_{2}=(12.0 \mathrm{cm}) \sin \left(4.5 t+70^{\circ}\right) $$ Find the amplitude and phase constant of the sum of these functions (a) by using a trigonometric identity (as from Appendix B) and (b) by representing the oscillations as phasors. State the result of comparing the answers to (a) and (b). (c) Phasors make it equally easy to add traveling waves. Find the amplitude and phase constant of the sum of the three waves represented by $$ \begin{array}{l}{y_{1}=(12.0 \mathrm{cm}) \sin \left(15 x-4.5 t+70^{\circ}\right)} \\ {y_{2}=(15.5 \mathrm{cm}) \sin \left(15 x-4.5 t-80^{\circ}\right)} \\ {y_{3}=(17.0 \mathrm{cm}) \sin \left(15 x-4.5 t+160^{\circ}\right)}\end{array} $$

O Marie Cornu, a physicist at the Polytechnic Institute in Paris, invented phasors in about 1880 . This problem helps you see their general utility in representing oscillations. Two mechanical vibrations are represented by the expressions
$$
y_{1}=(12.0 \mathrm{cm}) \sin (4.5 t)
$$
and
$$
y_{2}=(12.0 \mathrm{cm}) \sin \left(4.5 t+70^{\circ}\right)
$$
Find the amplitude and phase constant of the sum of these functions (a) by using a trigonometric identity (as from Appendix B) and (b) by representing the oscillations as phasors. State the result of comparing the answers to (a) and (b). (c) Phasors make it equally easy to add traveling waves. Find the amplitude and phase constant of the sum of the three waves represented by
$$
\begin{array}{l}{y_{1}=(12.0 \mathrm{cm}) \sin \left(15 x-4.5 t+70^{\circ}\right)} \\ {y_{2}=(15.5 \mathrm{cm}) \sin \left(15 x-4.5 t-80^{\circ}\right)} \\ {y_{3}=(17.0 \mathrm{cm}) \sin \left(15 x-4.5 t+160^{\circ}\right)}\end{array}
$$

Key Concepts

Sinusoidal Oscillations

Sinusoidal oscillations are functions that can be described by sine or cosine functions and represent periodic motions. They are fundamental in describing phenomena ranging from mechanical vibrations to electrical signals, where the amplitude, frequency, and phase constant define the oscillatory behavior.

Phasor Representation

Phasor representation involves expressing sinusoidal functions as rotating vectors (phasors) in the complex plane. This method simplifies the addition and subtraction of oscillations by transforming trigonometric functions into complex numbers, making operations like superposition more straightforward.

Trigonometric Identities for Wave Addition

Trigonometric identities, particularly the sine and cosine angle addition formulas, are used to combine sinusoidal functions algebraically. These identities allow one to rewrite the sum of sinusoidal functions as a single sinusoid with an overall amplitude and phase shift.

Vector Addition in the Complex Plane

The process of vector addition in the complex plane is central to phasor methods. Each sinusoidal function is represented as a vector (or complex number), and adding them involves summing their real and imaginary components, leading to a resultant vector that encapsulates the combined amplitude and phase.

Superposition Principle for Waves

The superposition principle states that when two or more waves overlap, the resultant wave is the sum of the individual waves. This concept applies to both standing and traveling waves and is fundamental in understanding interference and the resultant amplitude and phase when multiple waves interact.

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