Two travelling waves of equal amplitudes and equal frequencies move in opposite directions along a string. They interfere to produce a stationary wave whose equation is given by y = (10 cos π x sin (2π t)/T) cm
The amplitude of the particle at x = 4/3 cm will be ___________ cm.

Solution

Physics sir · 3 steps

In a standing wave, everything in front of the time part is the amplitude at that point, so just plug in x.

1. Split the equation. The wave is y = (10 cos π x)sin(2π t)/T. The sin(2π t)/T part makes each particle oscillate in time. The factor 10 cos π x fixes how far the particle at position x can move.

   A(x) = |10 cos π x| cm

2. Put in x. At x = 4/3 cm the angle is 4π/3, and cos 4π/3 = -1/2. So 10 cos π x = 10 × (-1/2) = -5 cm.
3. So. Amplitude is a size, so take the magnitude: A = |-5| = 5 cm.

The common slip: -5. Forgetting the modulus gives -5. cos 4π/3 is negative, but an amplitude is always positive.

The idea to keep. For y = A₀ cos kx sin ω t, the amplitude at position x is |A₀ cos kx|. A minus sign only means that point moves opposite in phase. It does not mean a negative amplitude.

Answer from the NTA final answer key. Solution by PrepWiser.