The displacement from equilibrium

Question

• The displacement from equilibrium caused by a wave on a string is given by y(x, t) = (-0.00185 m)sin[(42.8 m^-1)x - (744 s^-1)t]. For this wave, what are the following? (a) amplitude? (b) number of waves in 1 m? (c) number of complete cycles in 1 s? (d) wavelength? (e) speed? Ans:- 0.00185 m, 6.81 waves, 118 cycles, 0.147 m, 17.4 m/s.

• Point A in the figure is 30 cm below the ceiling. Determine how much longer it will take for a wave pulse to travel along wire 1 than along wire 2. Ans:-2.28

  

• Show that the function D = A ln(x + vt) is a solution of the wave equation,

(d²D/dt²) = υ²(d²D/dx²)

• A string with linear mass density of 0.1 kg/m is under a tension of 107 N. How much power must be supplied to the string to generate a sinusoidal wave of amplitude 3 cm and frequency 116 Hz? Ans:-782 W

• 15.42. A cowboy walks at a pace of about two steps per second, holding a glass of diameter 11 cm that contains milk. The milk sloshes higher and higher in the glass until it eventually starts to spill over the top. Determine the maximum speed of the waves in the milk. Ans:- 0.44 m/s

• A 49-cm-long wire with a mass of 10.8 g is under a tension of 55.5 N. Both ends of the wire are held rigidly while it is plucked. (a) What is the speed of the waves on the wire? (b) What is the fundamental frequency of the standing wave? (c) What is the frequency of the third harmonic? Ans:- 55.5 m/s, 51.2 Hz, 154 Hz

• Consider a guitar string stretching 84.6 cm between its anchored ends. The string is tuned to play middle C, with a frequency of 256 Hz, when oscillating in its fundamental mode, that is, with one antinode between the ends. If the string is displaced 1.84 mm at its midpoint and released to produce this note, what are the wave speed, v, and the maximum speed, V_max, of the midpoint of the string? Ans:- 433 m/s, 5.92 m/s

• The tension in a 2.9-m-long, 1.1-cm-diameter steel cable (ρ = 7800 kg/m³) is 810 N. What is the fundamental frequency of vibration of the cable? Ans:- 5.7 Hz