Question

A taut rope has a mass of 0.123 kg and a length of 3.54 m .What average power must be supplied to the rope to generate sinusoidal waves that have amplitude 0.200 mm and wavelength 0.600 mm if the waves are to travel at 22.0 m/s ?

Answer 1

Answer:

811.54 W

Explanation:

Solution

Begin with the equation of the time-averaged power of a sinusoidal wave on a string:

P = $\frac{1}{2\\}$ μ.T².ω².v

The amplitude is given, so we need to calculate the linear mass density of the rope, the angular frequency of the wave on the rope, and the frequency of the wave on the string.

We need to calculate the linear density to find the wave speed:

μ = $\frac{M}{L}$ = 0.123Kg/3.54m

The wave speed can be found using the linear mass density and the tension of the string:

v= 22.0 ms⁻¹

v = f/λ =  22.0/6.0×10⁻⁴

= 36666.67 s⁻¹

The angular frequency can be found from the frequency:

ω=  2πf=2π(36666.67s−1)  = 2.30 ×10⁻⁵s⁻¹

Calculate the time-averaged power:

P =$\frac{1}{2}$μΤ²×ω²×ν

= $\frac{1}{2}$ ×( 0.03475kg/m)×(0.0002)²×(2.30×10⁵)² × 22.0

= 811.54 W