A transverse harmonic wave travels on a rope according to the following expressi

Question

A transverse harmonic wave travels on a rope according to the following expression:

y(x,t) = 0.14sin(2.5x + 17t)

The mass density of the rope is = 0.113 kg/m. x and y are measured in meters and t in seconds.

Wave Equation 2 3 4 5 6789 10 11 A transverse harmonic wave travels on a rope according to the following expression: y(x,t) 0.14sin(2.5x 17t) The mass density of the rope is H 0.113 kg/m. x and y are measured in meters and t in seconds. 1) What is the amplitude of the wave? Submit 2) What is the frequency of oscillation of the wave? Hz Submit 3) What is the wavelength of the wave? m Submit 4) What is the speed of the wave? m/s Submit 5) What is the tension in the rope? N Submit 6) At x 3.8 m and t-0.43 s, what is the velocity of the rope? (watch your sign) m/s Submit 7) At x 3.8 m and t 0.43 s, what is the acceleration of the rope? (watch your sign) m/2 Submit

Explanation / Answer

Comparing with the wave equation, y = A sin(kx + t)

1) Amplitude, A = 0.14 m

2) Freuency, f = /2 = 17 / 2 = 2.71 Hz

3) Wavelength, = 2/k = 2 / 2.5 = 2.51 m

4) Wave speed, v = f = 2.71 * 2.51 = 6.80 m/s

5) Wave speed, v = (T/)1/2

=> T = v2 = 0.113 * 6.802 = 5.23 N

6) y = 0.14 sin(2.5x + 17t)

Differentiating with respect to time,

v = dy/dt = (0.14 * 17) cos(2.5x + 17t)

=> v = 2.38 cos(2.5x + 17t)

At x = 3.8 m, t = 0.43 s,

v = 2.38 * cos[(2.5 * 3.8) + (17 * 0.43)] = -1.08 m/s

7) a = dv/dt = -(2.38 * 17) sin(2.5x + 17t)

At x = 3.8 m, t = 0.43 s,

a = -(2.38 * 17) * sin[(2.5 * 3.8) + (17 * 0.43)] = 36.1 m/s2

8) In one oscillation, any small section of the rope moves from the mean position to extreme positions and back to the mean posiiton, hence the distance covered by the section of rope is 4A,

Average speed in one oscillation = Total distance travelled in one oscillation / Time period of oscillation

=> vavg = 4A/T = 4Af = 4 * 0.14 * 2.71 = 1.52 m/s

9) In the wave equation y = A sin(kx + t), since kx and t both have positive signs before them, the wave is moving in the -x direction.

10) Increasing the wavelength has no effect on the time period of oscillation.

11) Complete this part based on your understanding of this problem

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