A wave with a frequency of 200 Hz and an amplitude of 1 mm travels on a long string. The string has a linear mass density of 6 g m ⁻¹ and is subjected to a tension of 60 N . Determine the average power transmitted across a given point on the string.

Step-by-step solution

Given: Frequency f = 200 Hz Amplitude A = 1 mm = 10⁻³ m Linear mass density = 6 g m ⁻¹ = 6 10⁻³ kg m ⁻¹ Tension T = 60 N The wave speed v is given by: v = T = 60 6 10⁻³ = 10000 = 100 m s ⁻¹ The angular frequency is: = 2 f = 2 200 = 400 rad s ⁻¹ The average power transmitted across a given point on the string is: P_ avg = 1 2 v ^2 A^2 Substituting the values: P_ avg = 1 2 (6 10⁻³) 100 (400 )^2 (10⁻³)^2 P_ avg = 0.3 160000 ^2 10⁻⁶ P_ avg = 0.048 ^2 Using ^2 9.87 : P_ avg 0.048 9.87 0.47 W