Question

Part 1: A rope has one end tied to a vertical support. You hold the other end so that the rope is horizontal. If you move the end of the rope back and forth with a frequency of 4 Hz, the transverse wave you produce has a wavelength of 0.5 m. What is the speed of the wave in the rope?a. 0.13 m/sb. 8 m/sc. 2 m/sd. 4 m/sPart 2: A rope with a mass density of 1 kg/m has one end tied to a vertical support. You hold the other end so that the rope is horizontal and has a tension of 4 N. If you move the end of the rope back and forth, you produce a transverse wave in the rope with a wave speed of 2 m/s. If you double the amount of tension you exert on the rope, what is the wave speed?a. 2.8 m/sb. 1.0 m/sc. 2.0 m/sd. 0.25 m/se. 4.0 m/s

Answer 1

1) c. 2 m/s

Explanation:

The relationship between frequency, wavelength and speed of a wave is

$v=\lambda f$

where

v is the speed

$\lambda$ is the wavelength

f is the frequency

For the wave in this problem,

f = 4 Hz

$\lambda=0.5 m$

So, the speed is

$v=(0.5 m)(4 Hz)=2 m/s$

2)  a. 2.8 m/s

The speed of the wave on a string is given

$v=\sqrt{\frac{T}{\mu}}$

where

T is the tension in the string

$\mu$ is the linear mass density

In this problem, we have:

$T=2 \cdot 4 N=8 N$ (final tension in the rope, which is twice the initial tension)

$\mu = 1 kg/m$ --> mass density of the rope

Substituting into the formula, we find

$v=\sqrt{\frac{8 N}{1 kg/m}}=2.8 m/s$