Question

A transverse wave on a long horizontal rope with a
wavelength of 8 m travels at 2 m/s. At t= 0, a
particular point on the rope has a vertical
displacement of +A, where A is the amplitude of
the wave. At what time will the vertical
displacement of this same point on the rope be -A ?

Answer 1

Answer:

2 seconds

Explanation:

The frequency of a wave is related to its wavelength and  speed by the equation

$f=\frac{v}{\lambda}$

where

f is the frequency

v is the speed of the wave

$\lambda$ is the wavelength

For the wave in this problem,

v = 2 m/s

$\lambda=8 m$

So the frequency is

$f=\frac{2}{8}=0.25 Hz$

The period of a wave is equal to the reciprocal of the frequency, so for this wave:

$T=\frac{1}{f}=\frac{1}{0.25}=4 s$

This means that the wave takes 4 seconds to complete one full cycle.

Therefore, the time taken for the wave to go from a point with displacement +A to a point with displacement -A is half the period, therefore for this wave:

$t=\frac{T}{2}=\frac{4}{2}=2 s$

Answer 2

The time taken by the wave to travel the vertical  displacement from the point on the rope to -A is 2 seconds

 

Given to us:

wavelength λ = 8 m,

Velocity v = 2 m/s,

The frequency of a wave is given by:

$frequency=\dfrac{Velocity}{wavelength}\\f=\dfrac{V}{\lambda}\\\\f=\dfrac{2}{8}\\\\f= 0.25\ \rm Hz$

Therefore, the frequency of the wave is 0.25 Hz.

The period of a wave is equal to the reciprocal of the frequency,

${\rm period\ of\ time}(T),\\T=\dfrac{1}{f}\\T=\dfrac{1}{0.25}\\\\T=4\ sec$

Therefore, It takes 4 seconds to complete one full cycle.

Hence, the time taken by the wave to travel the vertical  displacement from the point on the rope to -A is 2 seconds as its is the half distance which is needed to be traveled by the wave.  

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