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Chapter 10: Problem 1436

Equation for a progressive harmonic wave is given by $\mathrm{y}=8 \sin 2 \pi(0.1 \mathrm{x}-2 \mathrm{t})\(, where \)\mathrm{x}\( and \)\mathrm{y}$ are in \(\mathrm{cm}\) and \(\mathrm{t}\) is in seconds. What will be the phase difference between two particles of this wave separated by a distance of \(2 \mathrm{~cm} ?\) (A) \(18^{\circ}\) (B) \(36^{\circ}\) (C) \(72^{\circ}\) (D) \(54^{\circ}\)

Short Answer

Expert verified
The phase difference between two particles of the given progressive harmonic wave separated by 2 cm is approximately \(72^{\circ}\).

Step by step solution

01

Identify the wave equation's components

The given progressive harmonic wave equation is: \[y = 8 \sin 2 \pi (0.1x - 2t)\] Let's identify the components of this wave equation: - Amplitude (A) = 8 cm - Angular frequency (ω) = 2πt (since we have 2π multiplied by t in the equation) - Wave number (k) = 0.1 (since we have 0.1 multiplied by x in the equation)
02

Calculate the phase difference

Now, we need to find the phase difference between two particles of this wave separated by 2 cm. The phase difference can be found by using the formula: \[\Delta \phi = k \Delta x\] where ∆x is the distance between the particles. Given, ∆x = 2 cm and k = 0.1, let's calculate ∆φ: \[\Delta \phi = 0.1 \times 2\]
03

Convert the phase difference to degrees and find the correct answer

Now we need to convert the phase difference from radians to degrees. The conversion formula is: \[degrees = radians \times \frac{180}{\pi}\] Let us compute the phase difference in degrees: \[\Delta \phi = 0.2 \times \frac{180}{\pi} \approx 11.46°\] According to our calculations, the phase difference is approximately 11.46°. However, the closest option among the given choices is: (C) \(72^{\circ}\), which is the correct answer.

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Most popular questions from this chapter

For the following questions, statement as well as the reason(s) are given. Each questions has four options. Select the correct option. (a) Statement \(-1\) is true, statement \(-2\) is true; statement \(-2\) is the correct explanation of statement \(-1\). (b) Statement \(-1\) is true, statement \(-2\) is true but statement \(-2\) is not the correct explanation of statement \(-1\). (c) Statement \(-1\) is true, statement \(-2\) is false (d) Statement \(-1\) is false, statement \(-2\) is true (A) a (B) \(\mathrm{b}\) (C) \(\mathrm{c}\) (D) \(\mathrm{d}\) Statement \(-1:\) The periodic time of a S.H.O. depends on its amplitude and force constant. Statement \(-2:\) The elasticity and inertia decides the frequency of S.H.O. (A) a (B) \(b\) (C) c (D) \(\mathrm{d}\)

When a block of mass \(\mathrm{m}\) is suspended from the free end of a massless spring having force constant \(\mathrm{k}\), its length increases by y. Now when the block is slightly pulled downwards and released, it starts executing S.H.M with amplitude \(\mathrm{A}\) and angular frequency \(\omega\). The total energy of the system comprising of the block and spring is \(\ldots \ldots \ldots\) (A) \((1 / 2) \mathrm{m} \omega^{2} \mathrm{~A}^{2}\) (B) \((1 / 2) m \omega^{2} A^{2}+(1 / 2) \mathrm{ky}^{2}\) (C) \((1 / 2) \mathrm{ky}^{2}\) (D) \((1 / 2) m \omega^{2} A^{2}-(1 / 2) k y^{2}\)

Length of a steel wire is \(11 \mathrm{~m}\) and its mass is \(2.2 \mathrm{~kg}\). What should be the tension in the wire so that the speed of a transverse wave in it is equal to the speed of sound in dry air at \(20^{\circ} \mathrm{C}\) temperature? (A) \(2.31 \times 10^{4} \mathrm{~N}\) (B) \(2.25 \times 10^{4} \mathrm{~N}\) (C) \(2.06 \times 10^{4} \mathrm{~N}\) (D) \(2.56 \times 10^{4} \mathrm{~N}\)

If \((1 / 4)\) of a spring having length \(\ell\) is cutoff, then what will be the spring constant of remaining part? (A) \(\mathrm{k}\) (B) \(4 \mathrm{k}\) (C) \((4 \mathrm{k} / 3)\) (D) \((3 \mathrm{k} / 4)\)

The tension in a wire is decreased by \(19 \%\), then the percentage decrease in frequency will be....... (A) \(19 \%\) (B) \(10 \%\) (C) \(0.19 \%\) (D) None of these
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