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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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Most popular questions from this chapter

The displacement from equilibrium caused by a wave on a string is given by \(y(x, t)=(-0.00200 \mathrm{~m}) \sin \left[\left(40.0 \mathrm{~m}^{-1}\right) x-\right.\) \(\left.\left(800 . \mathrm{s}^{-1}\right) t\right] .\) For this wave, what are the (a) amplitude, (b) number of waves in \(1.00 \mathrm{~m},\) (c) number of complete cycles in \(1.00 \mathrm{~s},\) (d) wavelength, and (e) speed?

A sinusoidal wave traveling in the positive \(x\) -direction has a wavelength of \(12 \mathrm{~cm},\) a frequency of \(10.0 \mathrm{~Hz},\) and an amplitude of \(10.0 \mathrm{~cm}\). The part of the wave that is at the origin at \(t=0\) has a vertical displacement of \(5.00 \mathrm{~cm} .\) For this wave, determine the a) wave number, d) speed, b) period, e) phase angle, and c) angular frequency, f) equation of motion.

A cowboy walks at a pace of about two steps per second, holding a glass of diameter \(10.0 \mathrm{~cm}\) that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk.

Why do circular water waves on the surface of a pond decrease in amplitude as they travel away from the source?

What is the wave speed along a brass wire with a radius of \(0.500 \mathrm{~mm}\) stretched at a tension of \(125 \mathrm{~N}\) ? The density of brass is \(8.60 \cdot 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\).
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