Question

A transverse traveling wave on a taut wire has an amplitude of 0.200 mm and a frequency of 570 hz. it travels with a speed of 196 m/s. (a) if the wave equation is written in the form y = a sin(kx - t), what are the parameters a, k, and ? m rad/m rad/s (b) the mass per unit length of this wire is 4.50 g/m. find the tension in the wire.

Answer 1

Step-by-step explanation:

(a) The wave equation for a transverse wave is given by y = a * sin(kx - ωt), where:

- y is the displacement of the wave

- a is the amplitude

- k is the wave number

- x is the position coordinate

- ω is the angular frequency

- t is time

Given:

Amplitude (a) = 0.200 mm = 0.0002 m

Frequency (f) = 570 Hz

Speed (v) = 196 m/s

Angular frequency (ω) is related to frequency by ω = 2πf.

So, ω = 2π * 570 rad/s = 3594π rad/s

The wave number (k) is related to the speed and angular frequency by k = ω/v.

k = (3594π rad/s) / (196 m/s) ≈ 18.283 rad/m

The parameters a, k, and ω are:

a = 0.0002 m

k ≈ 18.283 rad/m

ω ≈ 3594π rad/s

(b) The tension (T) in the wire is related to the wave speed (v) and the mass per unit length (μ) by the equation:

v² = T / μ

Given mass per unit length (μ) = 4.50 g/m = 0.0045 kg/m

Speed (v) = 196 m/s

T = μ * v² = 0.0045 kg/m * (196 m/s)² ≈ 176.904 N

So, the tension in the wire is approximately 176.904 Newtons.