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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

Two simple pendulums having lengths \(144 \mathrm{~cm}\) and \(121 \mathrm{~cm}\) starts executing oscillations. At some time, both bobs of the pendulum are at the equilibrium positions and in same phase. After how many oscillations of the shorter pendulum will both the bob's pass through the equilibrium position and will have same phase ? (A) 11 (B) 12 (C) 21 (D) 20

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