The minimum possible length of the string when both ends of...

The minimum possible length of the string when both ends of string are fixed and has consecutive standing wave mode for which distance between adjacent nodes are $18 \mathrm{~cm}$ and $16 \mathrm{~cm}$ respectively, is : (a) $150 \mathrm{~cm}$ (b) $144 \mathrm{~cm}$ (c) $140 \mathrm{~cm}$ (d) $142 \mathrm{~cm}$

The minimum possible length of the string when both ends of string are fixed and has consecutive standing wave mode for which distance between adjacent nodes are $18 \mathrm{~cm}$ and $16 \mathrm{~cm}$ respectively, is :
(a) $150 \mathrm{~cm}$
(b) $144 \mathrm{~cm}$
(c) $140 \mathrm{~cm}$
(d) $142 \mathrm{~cm}$

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Key Concepts

Standing Waves

Standing waves are formed by the superposition of two waves traveling in opposite directions, resulting in fixed points of no displacement called nodes. In strings with fixed ends, the boundary conditions force the displacement to be zero at the ends, leading to distinct harmonic modes.

Nodes and Antinodes

Nodes are points along a standing wave where the medium does not move due to destructive interference; the distance between adjacent nodes is half the wavelength. This spacing is crucial in determining the conditions for standing waves in a fixed boundary system.

Harmonics and Mode Numbers

In a string fixed at both ends, standing waves occur at discrete frequencies, each corresponding to a mode number or harmonic. The nth harmonic has n antinodes and n+1 nodes, and the mode number directly relates the string length to the wavelength.

Relationship Between String Length and Wavelength

For a string with fixed ends, the wavelength of the standing wave is determined by the length of the string and the harmonic mode. In the nth mode, the wavelength is given by ? = 2L/n. The distance between consecutive nodes, being half of the wavelength, provides a direct relationship that can be used to set up equations to determine unknown parameters like the string length.

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