speed of transverse waves on a string under tension & power

Question

(a) Determine the speed of transverse waves on a string under tension of 80.0 N if the string has a length of 2.00 m and a mass of 5.00 g. (b) Calculate the power required to generate these waves if they have a wavelength of 16.0 cm, and amplitude of 4.00 cm.

Answer 1

Hii,

To determine the speed of transverse waves on a string, we use the formula:

v = sqrt(T / μ)

where:

v = wave speed

T = tension in the string (80.0 N)

μ = linear mass density (mass per unit length)

(a) First, calculate the linear mass density:

μ = mass / length = (5.00 g) / (2.00 m) = (0.00500 kg) / (2.00 m) = 0.00250 kg/m

Now, calculate the wave speed:

v = sqrt(80.0 N / 0.00250 kg/m)

v = sqrt(32000)

v = 179 m/s

(b) The power required to generate the waves is given by:

P = (1/2) μ ω² A² v

where:

P = power

μ = 0.00250 kg/m

ω = angular frequency = 2πf

A = amplitude = 4.00 cm = 0.0400 m

v = 179 m/s

First, find the frequency using the wave equation:

v = f λ

f = v / λ = (179 m/s) / (0.160 m) = 1119 Hz

Now, calculate angular frequency:

ω = 2πf = 2π (1119) = 7032 rad/s

Now, calculate power:

P = (1/2) (0.00250) (7032)² (0.0400)² (179)

P = (0.00125) (49448624) (0.0016) (179)

P = (0.00125) (141623.8)

P = 177 W

Final answers:

(a) Wave speed = 179 m/s

(b) Power required = 177 W