Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

A sinusoidal transverse wave of wavelength 20cm travels along a string in the positive direction of an axis. The displacement y of the string particle at x=0 is given in Figure 16-34 as a function of time t. The scale of the vertical axis is set by $y_{s} = 4.0 cm$ The wave equation is to be in the form $y (x, t) = y_{m} \sin (kx \pm ωt + ϕ)$ . (a) At t=0, is a plot of y versus x in the shape of a positive sine function or a negative sine function? (b) What is $y_{m}$ , (c) What is k,(d) What is $ω$ , (e) What is $φ$ (f) What is the sign in front of $ω$ , and (g) What is the speed of the wave? (h) What is the transverse velocity of the particle at x=0 when t=5.0 s?

a) At t=0, a plot of y vs x in the slope of a negative sine function as: $y (x, 0) = - y_{m} s i n (k x)$ .

b) The amplitude $y_{m}$ is 4.0 cm.

c) The angular wave number k is,0.31rad/cm .

d) The angular frequency $ω$ is,0.63 rad/s .

e) The phase constant $ϕ$ is, $π$ .

f) The sign in front of $ω$ is, negative.

g) The speed of the wave v is,2.0 cm/s.

h) The transverse velocity of the particle at x=0 when t=5.0 s is,-2.5 cm/s .

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The given data

The wavelength of the wave, $λ = 20 cm$ .
The scale of the vertical axis is set by $y_{s} = 40 c m$ .
The wave equation is to be in the form, $y (x, t) = y_{m} s i n (k x \pm ω t + ϕ)$ .

Step 2: Understanding the concept of wave equation

By using a general expression for a sinusoidal wave traveling along the +x direction and corresponding formulas, we can find the amplitude, angular wave number k, angular frequency $ω$ , the phase constant $ϕ$ , the sign in front of $ω$ , and the speed of the wave v and the transverse velocity of the particle at x=0 when t=5.0 s.

Formula:

A general expression for a sinusoidal wave traveling along the +x direction,

$y (x, t) = y_{m} s i n (k x \pm ω t + ϕ)$ (i)

The angular wave number, $k = \frac{2 π}{λ}$ (ii)

The angular frequency, $ω = \frac{2 π}{T}$ (iii)

The frequency, $f = \frac{1}{T}$ (iv)

The speed of the wave, $v = f λ$ (v)

The transverse velocity of the particle, $u (x, t) = \frac{\partial y}{\partial t}$ (vi)

Step 3: a) Plotting y versus x graph

A general expression for a sinusoidal wave traveling along the direction using equation (vi) is given as:

$y (x, t) = y_{m} s i n (k x \pm ω t + ϕ)$

Figure 16-34 shows that at x=0

$y (0, t) = y_{m} s i n (- ω t + ϕ) \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . (1)$

And it is a positive sine function. That is

$y (0, t) = + y_{m} s i n (ω t)$

For the sin function, we can write that

From equation (1) and (2), we can say that the phase constant must be

$ϕ = π$

At t = 0, we have

$y (x, 0) = y_{m} s i n (k x + π)$

Using equation (2), we get the displacement equation as:

$y (x, 0) = - y_{m} s i n (k x)$ .

which is a negative sine function. A plot of $y (x, 0)$ is plotted below.

Step 4: b) Calculation for amplitude

From the figure we see that the amplitude is

$y_{m} = 4.0 cm$ .

Hence, the value of amplitude of the function is 4.0 cm.

Step 5: c) Calculation for the wavenumber

Using equation (ii) and the given value of wavelength, the angular wave number is given by:

$\begin{array}{l} k = \frac{2 π}{2 π} \\ = 0.31 rad / cm \end{array}$

Hence, the value of wavenumber is 0.31 rad/cm.

Step 6: d) Calculation for the angular frequency

Using equation (iii), the angular frequency is given by:

$ω = \frac{2 π}{10} = 0.63 rad / s$

Hence, the value of the angular frequency is 0.63 rad/s.

Step 7: e) Calculation for the phase constant

The figure shows that at x=0,

$y (0, t) = y_{m} s i n (- ω t + ϕ)$

And it is a positive sine function. That is

$y (0, t) = + y_{m} \sin (ωt)$

Therefore, the phase constant must be $ϕ = π$ .

Hence, the value of phase constant is $π$ .

Step 8: f) Finding the sign of angular frequency

The sign is minus since the wave is traveling in the +x direction. Hence, the sign of the angular frequency is negative.

Step 9: g) Calculation of the speed of the wave

Using equation (iv), the frequency of the wave is given as:

$f = \frac{1}{10} = 0.10 s^{- 1}$

Therefore, using equation (v) and the above value f frequency, the speed of the wave is given as:

$v = 0.10 \times 20 = 2.0 cm$

Hence, the value of speed of the wave is 2.0 cm/s.

Step 10: h) Calculation of the transverse velocity

From the results above, the wave may be expressed as

$y (x, t) = 4.0 \sin (\frac{πx}{10} - \frac{πt}{5} + π) = - 4.0 \sin (\frac{πx}{10} - \frac{πt}{5})$

Using the equation (vi ) and the above wave equation, the transverse velocity is given as:

$u (x, t) = \frac{d}{d t} (- 4.0 \sin (\frac{πx}{10} - \frac{πt}{5})) = - 4.0 (\frac{π}{t}) \cos (\frac{πx}{10} - \frac{πt}{5})$

Hence, at the required values, the value of transverses velocity is given by:

$u (0, 5, 0) = - 4.0 (\frac{π}{5.0}) \cos (- \frac{π \times 5.0}{5}) = - 2.5 cm / s$

Hence, the value of transverse velocity is 2.5 cm/s.

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of wave equation

Step 3: a) Plotting y versus x graph

Step 4: b) Calculation for amplitude

Step 5: c) Calculation for the wavenumber

Step 6: d) Calculation for the angular frequency

Step 7: e) Calculation for the phase constant

Step 8: f) Finding the sign of angular frequency

Step 9: g) Calculation of the speed of the wave

Step 10: h) Calculation of the transverse velocity

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Fundamentals Of Physics

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Short Answer

Step by Step Solution

Step 1: The given data

Step 2: Understanding the concept of wave equation

Step 3: a) Plotting y versus x graph

Step 4: b) Calculation for amplitude

Step 5: c) Calculation for the wavenumber

Step 6: d) Calculation for the angular frequency

Step 7: e) Calculation for the phase constant

Step 8: f) Finding the sign of angular frequency

Step 9: g) Calculation of the speed of the wave

Step 10: h) Calculation of the transverse velocity

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