Question

A tuning fork's frequency is f = 140 Hz, and its amplitude of oscillation (of each tine in the fork) is 0.60 mm. Transverse waves are created in a stretched wire by attaching it to this tuning fork, where the fork's motion is perpendicular to the wire, and parallel to gravity. The wire's linear mass density is p = 0.018 kg/m and it is under a tension of T = 1.2 kN. At t = 0, the fork is in its equilibrium position. Find each of the following quantities:

(a) The propagation speed, the wavelength, and the wave number.
(b) The wavefunction that describes this wave (with numerical physical quantities included).
(c) The maximum transverse speed and transverse acceleration of a point on the wire.
(d) The average rate at which energy must be supplied to the fork to keep it oscillating at a steady amplitude.

Answer 1

The propagation speed is 109.54 m/s, the wavelength is 0.782 m, and the wave number is 8.04 rad/m.

Step-by-step explanation:

To find the propagation speed of the wave on the wire, we can use the formula v = sqrt(T/μ), where T is the tension in the wire and μ is the linear mass density of the wire.

Using the given values of T = 1.2 kN and μ = 0.018 kg/m, we can calculate v = sqrt((1.2 x 10^3 N) / (0.018 kg/m)) = 109.54 m/s.

The wavelength can be found using the formula λ = v/f, where f is the frequency of the tuning fork. Using the given value of f = 140 Hz, we can calculate λ = (109.54 m/s) / (140 Hz) = 0.782 m.

The wave number can be found using the formula k = 2π/λ. Substituting the value of λ, we get k = 2π / 0.782 m = 8.04 rad/m.