A 2 in long string fixed at both ends is set into vibrations in i… - Vidyadip

Question

A $2$ in long string fixed at both ends is set into vibrations in its first overtone. The wave speed on the string is $200m/s$ and thee amplitude is $0.5\ cm.$

Find the wavelength and the frequency.
Write the equation giving the displacement of different points as a function of time. Choose the $X-$axis along the string with the origin at one end and $t = 0$ at the instant when the point $x = 50\ cm$ has reached its maximum displacement.

✓

Answer

$\text{V}=200\text{m/s},\ 2\text{A}=0.5\text{m}$

The string is vibrating in its $1^{st}$ overtone

$\Rightarrow\lambda=1=2\text{m}$
$\Rightarrow\text{f}=\frac{\text{v}}{\lambda}=100\text{Hz}$

The stationary wave equation is given by

$\text{y}=2\text{A}\cos\frac{2\pi\text{x}}{\lambda}\sin\frac{2\pi\text{Vt}}{\lambda}$
$=(0.5\text{cm})\cos\big[(\pi\text{m}^{-1})\text{x}\big]\sin\big[(200\pi\text{s}^{-1})\text{t}\big]$

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