Answer to Question #102969 in Electrical Engineering for Promise Omiponle

Question

1. A transverse wave on a taut string is given by
y(x, t)=(0. 750 cm)cos[pi{(0.400 rad/cm)x + (250rad/s)t}]
[Note the 휋휋 inside the cosine function!!!]
a) What is the amplitude?
b) What is the period?
c) What is the (linear) frequency?
d) What is the wavelength?
e) What is the speed of propagation?
f) What is the direction of propagation (+x-direction or –x-direction)?
g) If the mass per unit length is 50.0 g/m, what is the tension in the string in newtons?
h) What is the displacement of a point on the rope located at x = 56.4 cm at time t = 0.115 s?

Answer 1

$$y(x, t)=(0. 750 \text{ cm})\text{cos}\big[\pi{(0.400 \text{ rad/cm})x + (250\text{ rad/s})t}\big].$$

a) The amplitude is 0.750 cm.

b) Note the angular frequency before time variable:

$$\omega=250\text{ rad/s},\\ \space\\ T=\frac{2\pi}{\omega}=0.025\text{ s}.$$

c) The frequency is

$$f=\frac{\omega}{2\pi}=29.8\text{ Hz}.$$

d) The wavelength can be found from the wave number:

$$k=0.4\pi\text{ rad/cm},\\ \space\\ \lambda=\frac{2\pi}{k}=5\text{ cm}.$$

e) The speed:

$$v=\frac{\omega}{k}=198.9\text{ cm/s}.$$

f) Since the wave number is positive, the direction is also positive.

g) The tension:

$$T=v^2\mu=1.99^2\cdot50\cdot10^{-3}=0.198\text{ N}.$$

h) Just substitute the values (substitute x in centimeters):

$$y(x, t)=0.75\text{cos}\big[0.4\pi\cdot56.4 + 250\cdot0.115\big]=-0.125\text{ cm}.$$