A transverse sinusoidal wave of amplitude $a,$ wavelength $\lambda$ and frequency $n$ is travelling on a stretched string. The maximum speed of any point on the string is $v/10,$ where $v$ is the speed of propagation of the wave. If $a = {10^{ - 3}}\,m$ and $v = 10\,m{s^{ - 1}}$, then $\lambda$ and $n$ are given by
IIT 1998, Medium
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(d) ${v_{\max }} = a\omega = \frac{v}{{10}} = \frac{{10}}{{10}} = m/sec$
$ \Rightarrow $$a\omega = a \times 2\pi n = 1$$ \Rightarrow $$n = \frac{{{{10}^3}}}{{2\pi }}$ $(\because a=10^{-3})$
Since $v = n\lambda \Rightarrow \lambda = \frac{v}{n} = \frac{{10}}{{{{10}^3}/2\pi }} = 2\pi \times {10^{ - 2}}\,m$
$ \Rightarrow $$a\omega = a \times 2\pi n = 1$$ \Rightarrow $$n = \frac{{{{10}^3}}}{{2\pi }}$ $(\because a=10^{-3})$
Since $v = n\lambda \Rightarrow \lambda = \frac{v}{n} = \frac{{10}}{{{{10}^3}/2\pi }} = 2\pi \times {10^{ - 2}}\,m$
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