Standing wave points execute SHM with displacement zero at quarter period

Steps:

• Frequency f = 250 Hz gives period T = 1/f = 0.004 s = 4 ms for SHM of each point.
• At t = 0, the string configuration shows maximum displacements, so point P is at its amplitude y(0) = -1.0 mm.
• Displacement of P follows y(t) = -1.0 \cos(2\pi t / T) mm, starting from negative maximum.
• Time t = 0.5 ms equals T/4, so phase 2\pi t / T = \pi/2, \cos(\pi/2) = 0, thus y = 0 mm.

Why B is correct:

• In SHM, displacement is zero at quarter period from extreme position, by the cosine function definition.

Why the others are wrong:

• A: -1.0 mm is the displacement at t = 0 or integer multiples of T.
• C: +0.5 mm does not match the phase advance of \pi/2 radians.
• D: +1.0 mm occurs at half period t = T/2, where \cos(\pi) = -1, so y = +1.0 mm.

Final answer: B