Question

1. A 200 Hz harmonic wave with an amplitude equal to 2 cm moves along a 40 m long string that has a mass of 120 grams and a tension of 50N.

1. Considering that there is no energy lost, find: the power transmitted past a given point on the string and total average energy on a 20-m long segment of the string
2. If the energy is lost in the process, the amplitude of the wave decreases as it travels along the string y=A(x)sin (kx-ωt), where the amplitude decreases exponentially with the distance A(x)=A₀e^-bxFind an expression of the power transported by the wave as a function of x (x>0).

PLEASE DO THE FOLLOWING:

1) Briefly state the problem (Given:). (0.25 points)

2) Briefly state the question (Find:). (0.25 points)

3) Draw a picture of the situation with information shown.; (0.5 points)

4) Specify the meaning of variables (example: “x is the distance from the top surface of

the liquid to the bottom of the window” or “P1 is the pressure at point (1) in my

diagram”)

Answer 1

A linear mass density is given by -

_$\mu$ = m / L = [(0.12 kg) / (40 m)]

_$\mu$ = 0.003 kg/m

We know that, v = _$\sqrt{}$T / _$\mu$

v = _$\sqrt{}$[(50 N) / (0.003 kg/m)]

v = _$\sqrt{}$16666.6 m²/s²

v = 129.09 m/s

a. Considering that there is no energy lost.

The average power transmitted past a given point on the string which will be given by -

P_avg = 2$\pi$²_$\mu$ v A² f²

where, A = amplitude of a wave = 0.02 m

f = frequency of a wave = 200 Hz

then, we get

P_avg = [2 (3.14)² (0.003 kg/m) (129.09 m/s) (0.02 m)² (200 Hz)²]

P_avg = 122.1 W

The total average energy on a 20 m long segment of the string which will be given by -

E_total = P_avg x t

E_total = (122.1 W) [(2 / 100) s]

E_total = [(122.1 W) (0.02 s)]

E_total = 2.44 J