Respuesta :
Answer:
A) The wave equation is defined as
[tex]y(x,t) = A\cos(kx + \omega t)=0.0275\cos(0.0041x + 6.2t)\\[/tex]
Using the wave equation we can deduce the wave number and the angular velocity. k = 0.0041 and ω = 6.2.
The time it takes for one complete wave pattern to go past a fisherman is period.
[tex]\omega = 2\pi f\\ f = 1/ T[/tex]
T = 1.01 s.
The horizontal distance the wave crest traveled in one period is
[tex]\lambda = 2\pi / k = 2\pi / 0.0041 = 1.53\times 10^3~m[/tex]
[tex]y(x = \lambda,t = T) = 0.0275\cos(0.0041*1.53*\10^3 + 6.2*1.01) = 0.0275~m[/tex]
B) The wave number, k = 0.0041 . The number of waves per second is the frequency, so f = 0.987.
C) A wave crest travels past the fisherman with the following speed
[tex]v = \lambda f = 1.53\times 10^3 * 0.987 = 1.51\times 10^3~m/s[/tex]
The maximum speed of the cork floater can be calculated as follows.
The velocity of the wave crest is the derivative of the position with respect to time.
[tex]v(x,t) = \frac{dy(x,t)}{dt} = -(6.2\times 0.0275)\sin(0.0041x + 6.2t)[/tex]
The maximum velocity can be found by setting the derivative of the velocity to zero.
[tex]\frac{dv_y(x,t)}{dt} = -(6.2)^2(0.0275)\cos(0.0041*1.53\times 10^3 + 6.2t) = 0[/tex]
In order this to be zero, cosine term must be equal to zero.
[tex]0.0041*1.53\times 10^3 + 6.2t = 5\pi /2\\t = 0.255~s[/tex]
The reason that cosine term is set to be 5π/2 is that time cannot be zero. For π/2 and 3π/2, t<0.
[tex]v(x=\lambda, t = 0.255) = -(6.2\times0.0275)\sin(0.0041\times 1.53\times 10^3 + 6.2\times 0.255) = -0.17~m/s[/tex]
(a) The time taken "1.013 s".
(b) Number of waves "0.987 Hz".
(c) Maximum speed "0.1750 m/s".
A further explanation is below.
Given:
- [tex]y(x,t) = (2.75 \ cm) Cos [(0.41 \ rad/cm)x+(6.20 \ rad/s)t][/tex]
(a)
The time taken will be:
→ [tex]T = \frac{2 \pi}{W}[/tex]
[tex]= \frac{2 \pi}{6.20}[/tex]
[tex]= 1.013 \ s[/tex]
The covered horizontal distance will be:
→ [tex]\lambda = \frac{2 \pi}{K}[/tex]
[tex]= \frac{2 \pi}{0.410}[/tex]
[tex]= 15.3 \ cm[/tex]
(b)
Wave number,
- [tex]K = 0.410 \ rad/cm[/tex]
The number of waves per second will be:
→ [tex]f = \frac{1}{T}[/tex]
[tex]= \frac{1}{1.013}[/tex]
[tex]= 0.987 \ Hz[/tex]
(c)
The speed in which the wave crest travel will be:
→ [tex]v = f \lambda[/tex]
[tex]= 15.3\times 0.987[/tex]
[tex]= 15.1 \ cm/s \ or \ 0.151 \ m/s[/tex]
and,
The maximum speed of the cork floater will be:
→ [tex]v_1 = AW[/tex]
[tex]=2.75\times 6.20[/tex]
[tex]= 0.1750 \ m/s[/tex]
Thus the above answers are correct.
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