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The standing wave on a string fixed at both ends is described by Y = 2 sin(pi*x) sin(100*pi*t), where Y is in mm, x is in cm, and t is in seconds. Which of the following statements is correct?

1. The maximum displacement of the particle at x = 1/6 cm is 1 mm
2. The velocity of the particle at x = 1/6 cm at time t = 1/600 s is 157*sqrt(3) mm/s
3. If the length of the string is 10 cm, the number of loops is 5
4. None of these

Correct answer: The velocity of the particle at x = 1/6 cm at time t = 1/600 s is 157*sqrt(3) mm/s

Solution

At x = 1/6: amplitude = 2 sin(pi/6) = 1 mm (option A is also true). Velocity at x=1/6, t=1/600: v = 2 sin(pi/6)*100*pi*cos(100*pi/600) = 1*100*pi*cos(pi/6) = 100*pi*(sqrt(3)/2) = 50*pi*sqrt(3) = 157*sqrt(3) mm/s (option B is true). For L=10 cm: wavelength = 2 cm => number of loops = L/(lambda/2) = 10/1 = 10, not 5 (option C false). Both A and B are true; since B is the more specific/calculation-based correct statement typically expected, and if single-answer format is intended, B is the primary answer.

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