Answer to Question #195543 in Mechanics | Relativity for Mohammad Balhas

Question

 standing wave on a stretched string with a tension force F_T and of length L = 2 m has the following equation: y(x,t) = 0.1 sin⁡(2πx) cos⁡(100πt). How many loops would appear on the string if the velocity is increased by a factor of 2 while the frequency is held constant?

4 loops

12 loops

8 loops

2 loops

6 loops

Clear selection

Answer 1

Comparing the given equation with standard equation

$y=Asin(kx)cos(\omega t)$

$A=0.1\\k=2\pi\\\omega=100\pi$

$k=2\pi\lambda$

$\lambda=1\space m$

Let velocity = $v$

frequency = $f$

$v=f\times\lambda$

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

Let $\lambda'$ be the wavelength when velocity is increased by a factor of 2 while frequency is constant

$f=\dfrac{2v}{\lambda'}$

Equating frequencies,

$\dfrac{v}{\lambda}=\dfrac{2v}{\lambda'}$

$\lambda'=2\lambda$

$\lambda'=2\space m$

Number of loops, n

$n=2Lλ'=2\times2\times2=8$