A sinusoidal transverse wave is traveling along a string in the negative directi
ID: 2222990 • Letter: A
Question
A sinusoidal transverse wave is traveling along a string in the negative direction of an x-axis. The figure shows a plot of the displacement as a function of position at time t=0; the scale of the y axis is set by ys = 4.0 cm. The string tension is 7.6 N, and its linear density is 52.78 g/m. A) Find the amplitude. B) Find the wavelength C) find the wave speed D) What is the period of the wave E) Find the maximum transverse speed of a particle in the string. F) If the wave is of the form y(x,t) = ym sin(kx
Explanation / Answer
y(x, t) = ym*sin(kx +/- wt + phi)
dy/dt = +/-w*ym*cos(kx +/- wt + phi)
Since the wave is moving along the positive x direction, the correct phase must be: (kx - wt + phi).
4.0 cm = 0.04 m = y(0,0) = ym*sin(phi)
Therefore ym = (0.04/sin(phi)) (m)
Since: 0 = dy/dt(0,0 ) = -w*ym*cos(phi), phi must be (+/-)(pi/2)
Thus, ym = 0.04/sin(phi) = +/- 0.04 (m). We take the + sign.
The maximum value of dy/dt(0,t):
16 m/s = w*ym = w*ym
Therefore w = 16/0.04 = 400 (rad/second)
a) f = w/(2*pi) = 63.66 (Hz)
b) WL = 2*pi/k = 2*pi/(w/c) = 2*pi*c/w
= 2*pi*80/400 = pi/2.5 = 1.2566 m
c) ym = 0.04 (m)
d) k = w/c = 400/80 = 5.0
e) w = 400 rad/sec
f) If we take ym as positive, then since y(0,t) = 0.04*sin(phi-400*t), then at t = 0, the displacement is 0.04*sin(+/- (pi/2). It has to be "+", so phi = pi/2.
g) It has to be minus in front of the w in the phase of the sinusoid. Why? Because you follow a wave by fixing the value of the phase, and then seeing how x changes at t changes. When constant = phase = kx - wt + phi
0 = d(phase)/dt = k*dx/dt - w
=> dx/dt = w/k , with the correct sign