Question

Waves on a string are described by the following general equation

y(x,t)=Acos(kx−ωt).

A transverse wave on a string is traveling in the +xdirection with a wave speed of 8.75 m/s , an amplitude of 6.50×10−2 m , and a wavelength of 0.540 m . At time t=0, the x=0 end of the string has its maximum upward displacement. Find the transverse displacement y of a particle at x = 1.59 m and t = 0.150 s .

Q. In general, the cosine function has maximum displacements, either positive or negative, when its argument is equal to an integer multiple of π. When t = 0.150 s , k = 11.6 rad/m , and ω = 102 rad/s use the wave equation to select all of the x positions that correspond to points of maximum displacement.

Check all that apply considering only positive arguments of the cosine function.

0.795 m
1.59 m
1.73 m
1.86 m
2.00 m
2.13 m

Answer 1

Answer:

The displacement is at x=1.59m and t=0.150s is $-6.50\cdot10^{-2}\text{ m}.$

Out of the given points, the argument of the cosine is an integer multiple of $\pi$ for x=1.59m, 1.86m, 2.13m.

Explanation:

The displacement at x and t is given by y(x,t). We now need to find $k$ and $\omega$. The speed of the wave is given by

$c=\frac{\omega}{k}$

while the wavelength satisfies

$k=\frac{2\pi}{\lambda}=\frac{2\pi}{0.540\text{ m}}=11.6\text{ m}^{-1}.$

Substituting this into the previous equation we find

$\omega=ck=8.75\text{ m/s}\cdot 11.6\text{ m}^{-1}=102\text{ rad/s}.$

Now we have

$y(1.59\text{ m},0.150\text{ s})=6.50\cdot10^{-2}\text{ m}\cos(11.6\cdot1.59-102\cdot0.150)=-6.50\cdot10^{-2}\text{ m}.$

Now we calculate $kx-\omega t$ at $t=0.150\text{ s}$ at each given x and check whether it is integer multiple of $\pi$.

$11.6\text{ m}^{-1}\cdot 0.795\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=-6.08=-1.93\pi\text{ not an integer multiple of }\pi;$

$11.6\text{ m}^{-1}\cdot 1.59\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=3.14=\pi\text{ it is an integer multiple of }\pi;$

$11.6\text{ m}^{-1}\cdot 1.73\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=4.77=1.52\pi\text{ not an integer multiple of }\pi;$

$11.6\text{ m}^{-1}\cdot 1.86\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=6.28=2\pi\text{ it is an integer multiple of }\pi;$

$11.6\text{ m}^{-1}\cdot 2.00\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=7.90=2.51\pi\text{ not an integer multiple of }\pi;$

$11.6\text{ m}^{-1}\cdot 2.13\text{ m}-102\text{ rad/s}\cdot0.150\text{ s}=9.41=3\pi\text{ it is an integer multiple of }\pi;$