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Chapter 16: Q39P (page 474)

Two sinusoidal waves of the same period, with amplitudes of 5.0and 7.0 mm, travel in the same direction along a stretched string; they produce a resultant wave with an amplitude of 9.0 mm. The phase constant of the 5.0 mmwave is 0.What is the phase constant of the 7.0 mmwave?

Short Answer

Expert verified

Phase constant of the 7 mm wave is $84 掳$

Step by step solution

01

The given data

i) Amplitudes of two waves, $y_{m 1} = 5.0 mm and y_{m 2} = 7.0 mm$

ii) Amplitude of resultant wave, $y_{m} = 9.0 mm$

iii) Phase constant of 5.00 mm wave, $胃 = 0 掳$

02

Understanding the concept of phase

The phase constant of sinusoidal waves of the same period which travel in the same direction is given by the cosine formula, and it can be shown in a phase diagram.

Formula:

Phase constant using Cosine Formula,

$y_{m}^{2} = y_{m^{1}}^{2} + y_{m^{2}}^{2} - 2 y_{m 1} y_{m 2} 肠辞蝉胃 = y_{m^{1}}^{2} + y_{m^{2}}^{2} - 2 y_{m 1} y_{m 2} \cos 蠁$ (i)

03

Calculation of phase of 7.00 mm wave

Phasor diagram can be drawn as below:

As we need to find the phase constant i.e. $蠁$ Hence, using equation (i) and given values, we get the value of phase as:

$\cos 蠁 = \frac{y_{m}^{2} - y_{m 1}^{2} + y_{m 2}^{2}}{2 y_{m 1} y_{m 2}} \cos 蠁 = \frac{{(9.0)}^{2} - {(5.0)}^{2} + {(7.0)}^{2}}{2 (5.0) (7.0)} \cos 蠁 = 0.10 蠁 = 84 掳$

Hence, the value of phase is $84 掳$

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Most popular questions from this chapter

(a) If a standing wave on a string is given by

$y (t) = (3 m m) s i n (5 x) c o s (4 t)$

is there a node or an antinode of the oscillations of the string atx = 0? (b) If the standing wave is given by

$y (t) = (3 mm) s i n (5 x + p / 2) c o s (4 t)$

is there a node or an antinode at x = 0?

A certain transverse sinusoidal wave of wavelength 20cmis moving in the positive direction of an xaxis. The transverse velocity of the particle at x = 0as a function of time is shown in Fig. 16-49, where the scale of the vertical axis is set by. What are the (a) wave speed, (b) amplitude, and (c) frequency? (d) Sketch the wave between x = 0and x = 20cm at t = 2.0 s.

Two waves are generated on a string of length 3.0 , to produce a three-loop standing wave with an amplitude of 1.0 cm. The wave speed is 100 m/s. Let the equation for one of the waves is of the form $y (x, t) = y_{m} s i n (k x + 蝇 t)$ . In the equation for the other wave, What are (a) $y_{m}$ (b)k, (c) What is $蝇$ , and (e) the sign in front of $蝇$ ?

A string under tension $蟿_{i}$ oscillates in the third harmonic at frequency $f_{3}$ , and the waves on the string have wavelength $位_{3}$ . If the tension is increased to $蟿_{f} = 4 蟿_{i}$ and the string is again made to oscillate in the third harmonic. What then are (a) the frequency of oscillation in terms of $f_{3}$ and (b) the wavelength of the waves in terms of $位_{3}$ ?

A string along which waves can travel is2.70 mlong and has a mass of 260 g. The tension in the string is 36.0 N. What must be the frequency of traveling waves of amplitude 7.70 mmfor the average power to be 85.0 W?

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