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Chapter 15: Problem 36

A sinusoidal wave on a string is described by the equation \(y=(0.100 \mathrm{~m}) \sin (0.75 x-40 t),\) where \(x\) and \(y\) are in meters and \(t\) is in seconds. If the linear mass density of the string is \(10 \mathrm{~g} / \mathrm{m}\), determine (a) the phase constant, (b) the phase of the wave at \(x=2.00 \mathrm{~cm}\) and \(t=0.100 \mathrm{~s}\) (c) the speed of the wave, (d) the wavelength, (e) the frequency, and (f) the power transmitted by the wave.

Short Answer

Expert verified
Answer: The phase constant is (a) ϕ = 0. The phase of the wave at x = 2.00 cm and t = 0.100 s is (b) Θ = -3.97 radians. The speed of the wave is (c) v = 53.33 m/s. The wavelength is (d) λ = 8.377 m. The frequency is (e) f = 6.366 Hz. The power transmitted by the wave is (f) P = 42.22 W.

Step by step solution

01

Identify the given parameters from the wave equation

The wave equation is given as \(y(x, t) = (0.100 m) \sin (0.75 x - 40 t)\). From this equation, we can identify the given parameters: Amplitude (A) = 0.100 m Wave number (k) = 0.75 rad/m Angular frequency (ω) = 40 rad/s
02

Calculate the phase constant (a)

The phase constant (ϕ) is not explicitly given in the equation. However, we can observe that the equation is in the standard form \(y(x, t) = A\sin(kx - \omega t + \phi)\), and the phase constant is 0. Therefore, the phase constant is: \(\phi = 0\).
03

Calculate the phase of the wave (b)

To find the phase of the wave at a given position (\(x = 2.00 cm\)) and time (\(t = 0.100 s\), we can use the equation \(\Theta = kx-\omega t+\phi\). Thus: \(\Theta = (0.75)(0.02) - (40)(0.100) + 0\) \(\Theta = -3.97\) radians
04

Calculate the speed of the wave (c)

To find the speed of the wave (v), we can use the relationship \(v=\frac{\omega}{k}\). Thus: \(v=\frac{40 \text{ rad/s}}{0.75\text{ rad/m}}=53.33 \frac{\text{m}}{\text{s}}\)
05

Calculate the wavelength (d)

We can find the wavelength (λ) by using the relationship \(\lambda = \frac{2\pi}{k}\). Thus: \(\lambda = \frac{2\pi}{0.75\text{ rad/m}}=8.377 \text{ m}\).
06

Calculate the frequency (e)

We can find the frequency (f) by using the relationship \(f=\frac{\omega}{2\pi}\). Thus: \(f = \frac{40\text{ rad/s}}{2\pi}=6.366 \text{ Hz}\).
07

Calculate the power transmitted by the wave (f)

Finally, we can find the power transmitted by the wave (P) using the relationship \(P=\frac{1}{2} \rho v \omega^2A^2\). The linear mass density of the string is given as \(\text{10 g/m}=\text{0.01 kg/m}\). Thus: \(P = \frac{1}{2}(0.01 \frac{\text{kg}}{\text{m}})(53.33 \frac{\text{m}}{\text{s}})(40 \text{ rad/s})^2(0.100 \text{ m})^2\) \(P = 42.22 \text{ W}\). So, the required values are: (a) \(\phi = 0\), (b) \(\Theta = -3.97\) radians, (c) \(v = 53.33 \frac{\text{m}}{\text{s}}\), (d) \(\lambda = 8.377 \text{ m}\), (e) \(f = 6.366 \text{ Hz}\), and (f) \(P = 42.22 \text{ W}\).

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