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A string is clamped at both the ends and it is vibrating in its 4th harmonic. The equation of the stationary wave is y = 0.3 sin(0.157x) cos(200πt). The length of the string is: (All quantities are in SI units.)

1. (1) 40 m
2. (2) 80 m
3. (3) 20 m
4. (4) 60 m

Correct answer: (2) 80 m

Solution

In the 4th harmonic of a clamped string, there are 4 segments (or loops) along the length of the string. The wavelength can be determined from the wave equation, where the wave number k = 0.157 m⁻¹ gives a wavelength of 2π/k. Since the length of the string is equal to 4 times the wavelength divided by 4 (for the 4 segments), the total length calculates to 80 m.

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