Question

A wave on a string is described by
D(x,t)=
(2.00cm)sin[(12.57rad/m)x?(638rad/s)t],
1. where x is in m and t is in s. The linear density of the string is 5.00g/m.
2. What is the string tension?
3. What is the maximum displacement of a point on the string?
4. What is the maximum speed of a point on the string?

Answer 1

Explanation:

A wave on a string is described is given by :

$D(x,t)=2\ cm\ sin[(12.57\ rad/m)-(638\ rad/s)t]$

The linear density of the string is 5 g/m.

Where

x is in meters and t is in seconds

The general equation of a wave is given by :

$y=A\ sin(kx-\omega t)$

(2) The speed of the wave in terms of tension is given by :

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

Also, $v=\dfrac{\omega}{k}$

So, $\dfrac{\omega}{k}=\sqrt{\dfrac{T}{\mu}}$

$T=\dfrac{\mu \omega^2}{k^2}$

$T=\dfrac{5\times 10^{-3}\times (638)^2}{(12.57)^2}$

T = 12.88 N

(3) The maximum displacement of a point on the string is equal to the amplitude of the wave. So, the maximum displacement is 2 cm.

(4) The maximum speed of a point on the string is given by :

$v=A\omega$

$v=0.02\times 638$

v = 12.76 m/s

Hence, this is the required solution.