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Standard XII

Physics

Damped Oscillations

A string fixe...

Question

A string fixed a both ends is vibrating in the lowest mode of vibration for which a point at quarter of its length from one end is a point of maximum displacement. The frequency of vibration in this mode is 100 Hz. What will be the frequency emitted (in Hz) when it vibrates in the next mode such that this point is again a point of maximum displacement?

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Solution

Let length equal to 'l'.
Now one fourth the distance 'l' corresponds to max displacement (for the lowest such mode).
We assume that it is in 2nd mode of vibration.
For which $l = λ$
i.e. $v = f λ = 100 l$
Now for the same condition but the next closest mode, we see that $λ = \frac{l}{4}$
In which case.
$v = 100 l = f \frac{l}{4}$
$f = 400 H z$

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Let length equal to 'l'.Now one fourth the distance 'l' corresponds to max displacement (for the lowest such mode).We assume that it is in 2nd mode of vibration.For which l=λi.e. v=fλ=100lNow for the same condition but the next closest mode, we see that λ=l4In which case.v=100l=fl4f=400Hz