Question

A long string carries a wave; a 7.00-m segment of the string contains five complete wavelengths and has a mass of 180 g. The string vibrates sinusoidally with a frequency of 45.0 Hz and a peak-to-valley displacement of 11.0 cm. (The "peak-to-valley" distance is the vertical distance from the farthest positive position to the farthest negative position.)

(a) Write the function that describes this wave traveling in the positive x direction. (Use the following as necessary: x and t. x is in meters and t is in seconds. Enter your numerical coefficients to four significant figures.)
y =

(b) Determine the power being supplied to the string.
W

Answer 1

Standard function for a wave travelling in +x-direction is given by:

y = A*cos (kx - wt)

A = Amplitude of wave = ?

given that peak-to peak displacement = 11.0 cm

A = peak-to-peak displacement/2 = 11.0 cm/2 = 5.5 cm = 0.055 m

w = Angular frequency = 2*pi*f

f = frequency = 45.0 Hz

w = 2*pi*45.0 = 90*pi = 282.7

k = wave number = 2*pi/lambda

lambda = wavelength of string

lambda = total length of string/number of complete wavelengths = 7.00 m/5 = 1.40 m

k = 2*pi/1.40 = 4.48798

k = 4.50

So,

y = 0.055*sin (4.50*x - 282.7*t)

Part B.

Power supplied to the string is given by:

P = (1/2)u*w^2*A^2*v

u = linear mass density = mass per unit length = total mass/total length

u = 180 gm/7.00 m = 0.0257 kg/m

w = 282.7 rad/sec

A = 0.055 m

v = speed of wave = w/k = 282.7/4.50 = 62.82 m/sec

So,

P = (1/2)*0.0257*282.7^2*0.055^2*62.82

P = 195.2 W

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