A tuning fork with a frequency of 440 Hz is connected to a long string having a linear mass density of 0.01 kg m ⁻¹ and maintained under a tension of 49 N . The tuning fork generates transverse waves with an amplitude of 0.50 mm along the string. Determine the average rate at which the tuning fork transmits energy to the string.

Step-by-step solution

Given: Frequency f = 440 Hz Linear mass density = 0.01 kg m ⁻¹ Tension T = 49 N Amplitude A = 0.50 mm = 0.5 10⁻³ m The wave speed v is given by: v = T = 49 0.01 = 70 m s ⁻¹ The angular frequency is: = 2 f = 2 440 = 880 rad s ⁻¹ The average rate at which energy is transmitted (average power) is: P_ avg = 1 2 v ^2 A^2 Substituting the values: P_ avg = 1 2 0.01 70 (880 )^2 (0.5 10⁻³)^2 P_ avg = 0.35 774400 ^2 0.25 10⁻⁶ P_ avg = 0.06776 ^2 Using ^2 9.87 : P_ avg 0.06776 9.87 0.67 W