A sinusoidal wave is traveling on a string with speed 41.3 cm/s. The displacemen
ID: 1558964 • Letter: A
Question
A sinusoidal wave is traveling on a string with speed 41.3 cm/s. The displacement of the particles of the string at x = 13 cm is found to vary with time according to the equation y = (4.2 cm) sin[2.3 - (7.3 s-1)t]. The linear density of the string is 4.7 g/cm. What are (a) the frequency and (b) the wavelength of the wave? If the wave equation is of the form y(x,t) = ym sin(kx - t), what are (c) ym, (d) k, and (e) , and (f) the correct choice of sign in front of ? (g) What is the tension in the string?
Explanation / Answer
a)
We know :
y(x,t) = ym sin(kx - t),
And in the question we have
y = (4.2 cm) sin[2.3 - (7.3 s-1)t]
Hence,
t = (7.3 s^-1)t
= (7.3 rad/s)
Since,
= 2pif
f = /(2pif)
f = 7.3/(2pi)
f = 1.162 Hz
b)wavelength
v = lambda*f
lambda = v/f
lambda = 41.3/1.162 = 35.542 cm/s = 0.355 m/s or 0.36 m/s
c)
ym = 4.2 cm
d)
kx = 2.3
k = 2.3/x
k = 2.3/13 = 0.1769 /cm or 0.18 /cm
e) = (7.3 rad/s)
f) to identify the direction of the wave we need to check the time component of the phase which is as below
t = 7.3 s1t,
Hence, this means that the wave is travelling in the positive x-directions.
So the sign must be negative, since both time and frequency must have positive values.
g)
tension
v = (tau/mu)^1/2
tau = muv^2
tau = 4.7*(41.3)^2 = 8016.743 = 0.0802 N