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At which three points x1, x2, and x3 closest to x=0 but with x>0 will the displacement of the string y(x,t) be zero for all times? These are the first three nodal points. Express the first three nonzero nodal points as multiples of the wavelength λ, using constants like π. List the factors that multiply λ in increasing order, separated by commas.

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danishmallick97

Answer:

x = Æ›/4, 3Æ›/4, 5Æ›/4

Step-by-step explanation:

The equation for the transverse wave displacement of a string is given as:

y(x,t) = Acos(kx)sin(wt) = A/2 ( sin(kx + wt) - sin(kx -wt))----------(1)

Since we are aware of the product rule:

cosAsinB = 1/2( sin(A+B) - sin(A - B))

Att = 0, y(x,0) = A/2 (sin(kx - sinkx) = 0

(c) The angular frequency w= 2Ï€/T

Substituting the angular frequency in equation (1) we get

y(x,T/4) = Acos(kx)sin ((2Ï€/T)(T/4))

            = Acos(kx)sin(π/2)

            =Acos(kx)

(d) The first three non-zero nodal points will be:

for y = 0; coskx = 0

kx = π/2, 3π/2, 5π/2

NOW,

k = 2Ï€/ Æ›

(2π/ ƛ)x = π/2, 3π/2, 5π/2

x = ( π/2, 3π/2, 5π,2)*ƛ/2π

x = Æ›/4, 3Æ›/4, 5Æ›/4

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