A string is stretched between fixed points separated by 75 cm . It is observed to have a resonan

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A string is stretched between fixed points separated by 75...

A string is stretched between fixed points separated by 75 $\mathrm{cm} .$ It is observed to have a resonant frequencies of $420 \mathrm{~Hz}$ and $315 \mathrm{~Hz}$. There are other resonant frequencies between these two. Then the lowest frequency for this string is $\ldots \ldots \ldots \ldots . . . h z$ (A) $1.05$ (B) 1050 (C) $10.5$ (D) 105

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Key Concepts

Standing Waves

Standing waves occur when waves of the same frequency and amplitude travel in opposite directions and interfere, producing fixed nodes and antinodes. In systems with fixed boundaries, such as a stretched string, only waves that form standing wave patterns are allowed, leading to discrete resonant frequencies.

Resonant Frequencies

Resonant frequencies are the specific frequencies at which a system naturally oscillates with maximum amplitude. For a string fixed at both ends, these frequencies are determined by the requirements of standing wave formation and occur only at certain discrete values.

Fundamental Frequency and Harmonics

The fundamental frequency is the lowest resonant frequency of a vibrating system, while harmonics are higher frequencies that are integer multiples of the fundamental. In a string fixed at both ends, the resonant frequencies follow the relationship that each is an integer multiple of the fundamental frequency, ensuring the formation of consistent standing wave patterns.

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