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 total energy associated with the wave in a 2.0 m long portion of 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 Length of the string, L = 2.0 m Tension, T = 60 N The wave speed is v = T = 60 6 10⁻³ = 100 m s ⁻¹ . The wavelength is = v f = 100 200 = 0.5 m . Since L = 2.0 m = 4 , the length is an integer multiple of the wavelength, so the average energy density formula can be used exactly. The total energy E of the wave in a length L is given by: E = 1 2 ^2 A^2 L Substituting = 2 f : E = 1 2 (2 f)^2 A^2 L E = 2 ^2 f^2 A^2