A transverse sinusoidal wave is generated at one end of a long, horizontal string by a bar that moves up and down through a distance of $$ 1.00 \mathrm{cm} . $$ The motion is continuous and is repeated regularly 120 times per second. The string has linear density 120 $$ \mathrm{g} / \mathrm{m} $$ and is kept under a tension of $$ 90.0 \mathrm{N} $$. Find the maximum value of the transverse speed $$ u $$ ?
Let the displacement of the string be of the form $$ y(x, t)=y_{m} \sin (k x-\omega t) $$.
The velocity of a point on the string is
$$u(x, t)=\partial y / \partial t=-\omega y_{m} \cos (k x-\omega t)$$
and its maximum value is $$ u_{m}=\omega y_{m} $$.
For this wave the frequency is $$ f=120 \mathrm{Hz} $$ and the angular frequency is $$ \omega=2 \pi f=2 \pi(120 \mathrm{Hz})=754 \mathrm{rad} / \mathrm{s} $$.
since the bar moves through a distance of $$ 1.00 \mathrm{cm}, $$ the amplitude is half of that, or $$ y_{m}=5.00 \times 10^{-3} \mathrm{m} . $$
The maximum speed is $$u_{m}=(754 \mathrm{rad} / \mathrm{s})\left(5.00 \times 10^{-3} \mathrm{m}\right)=3.77 \mathrm{m} / \mathrm{s}$$
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