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A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is 1.0 cm and the displacement becomes zero 200 times per second. The linear mass density of the string is 0.10 kg m-1 and it is kept under a tension of 90 N. (a) Find the speed and the wavelength of the wave. (b) Assume that the wave moves in the positive x-direction and at t = 0, the end x = 0 is at its positive extreme position. Write the wave equations. (c) Find the velocity and acceleration of the particle at x = 50 cm at time t = 10 ms.

A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is 1.0 cm and the displacement becomes zero 200 times per second. The linear mass density of the string is 0.10 kg m-1 and it is kept under a tension of 90 N. (a) Find the speed and the wavelength of the wave. (b) Assume that the wave moves in the positive x-direction and at t = 0, the end x = 0 is at its positive extreme position. Write the wave equations. (c) Find the velocity and acceleration of the particle at x = 50 cm at time t = 10 ms.

Grade:11

1 Answers

Jitender Pal
askIITians Faculty 365 Points
9 years ago
Sol. Amplitude, A = 1 cm, Tension T = 90 N Frequency, f = 200/2 = 100 Hz Mass per unit length, m = 0.1 kg/mt a) ⇒ V = √T /m = 30 m/s λ = V/f = 30/100 = 0.3 m = 30 cm b) The wave equation y = (1 cm) cos 2π (t/0.01 s) – (x/30 cm) [because at x = 0, displacement is maximum] c) y = 1 cos 2π(x/30 – t/0.01) ⇒ v = dy/dt = (1/0.01)2π sin 2π {(x/30) – (t/0.01)} a = dv/dt = – {4π2 / (0.01)2} cos 2π {(x/30) – (t/0.01)} When, x = 50 cm, t = 10 ms = 10 * 10–3 s x = (2π / 0.01) sin 2π {(5/3) – (0.01/0.01)} = (p/0.01) sin (2π * 2 / 3) = (1/0.01) sin (4π/3) = –200 π sin (π/3) = –200 πx (√ 3 / 2) = 544 cm/s = 5.4 m/s Similarly a = {4π2 / (0.01)2} cos 2π {(5/3) – 1} = 4π2 * 104 * ½ ⇒ 2 * 105 cm/s2 ⇒ 2 km/s2

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