For a certain transverse standing wave on a long string, an antinode is at x = 0

Question

For a certain transverse standing wave on a long string, an antinode is at x = 0 and an adjacent node is at x = 0.40 m. The displacement y(t) of the string particle at x = 0 is shown in the figure, where the scale of the y axis is set by ys = 4.4 cm. When t = 0.90 s, what is the displacement of the string particle at (a) x = 0.20 m and (b) x = 0.30 m ? What is the transverse velocity of the string particle at x = 0.20 m at (c) t = 0.90 s and (d) t = 1.0 s?

For a certain transverse standing wave on a long string, an antinode is at x-0 and an adjacent node is at x 0.40 m. The displacement y(t) of the string particle at x 0 is shown in the figure, where the scale of the y axis is set by ys = 4.4 cm. when t 0.90 s, what is the displacement of the string particle at (a) x = 0.20 m and (b) x = 0.30 m ? what is the transverse velocity of the string particle at x 0.20 m at (c) t = 090 s and (d) t: 10 s? 0s 0.72.1 2.8 (a) Number T4.154 (b) Number 3.285 (c) NumberTT4.154 (d) Numbero.0437 UnitšT m UnitsT m UnitsT m/s UnitsT m/s

Explanation / Answer

From the Problem, we can conclude that amplitude A (= ys) = 4.4 cm time period T = 2.8 s

Initial phase= 90 degrees

wavelength,= 4 * distance between consecutive antinode and node = 4 * 0.40 = 1.60 m

Angular frequency, = 2 / T = 2 * 3.14 / 2.80 = 2.24 rad/s

k = 2*pi / = 2*pi/1.6 = 1.25 pi

Standard equation of wave is

y(x,t) = A * Cos (kx)Sin (*t)

y(x.t) = -0.044 cos (1.25*pi*x) Sin (2.24*t)

Transverse speed vt = dy/dt = -0.044 ** cos (1.25*pi*x) Cos (2.24*t)

Now subsitute for the required times.

When t = 0.90 s, what is the displacement of the string particle at (a) x = 0.20 m and (b) x = 0.30 m

y(x,t) = -0.044 cos (1.25*pi*x) Sin (2.24*t)

y(0.2,0.9) = -0.044 cos (1.25*pi*0.2)* Sin(2.24*0.9) = -0.44*0.707*0.9025 = 0.02807 m

Similiarly for x = 0.3 m,

y(0.3,0.9) = -0.044 cos (1.25*pi*0.3)* Sin(2.24*0.9) = 0.0152 m

The Transverse Velocity is

v(x.t) =-0.044 *2.24* cos (1.25*pi*x) Cos(2.24*t)

at x = 0.2,

v(0.2.t) =-0.044 *2.24* cos (1.25*pi*x) Sin (2.24*t)

v(0.2,t) = -0.09856 * 0.707 * Cos (2.24*t)

v(0.2,t) = -0.06968 * Cos (2.24*t)

For, t = 0.9 seconds, V = -0.06968 * Cos (2.24*0.9) = 0.03 m/s

For t = 1 second, V = -0.06968 * cos (2.24* 1) = 0.04322 m/s

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