1. A transverse sinusoidal wave is traveling on a string. The displacement of th

Question

1. A transverse sinusoidal wave is traveling on a string. The displacement of the particles on the string is foundto vary as:y(t) = a sin(bx ct + 2)
Where a = 0.01 m, b = 3 m1 and c = 4 s1. The linear mass density of the string is 4 kg/m.
(a) What is the frequency and wavelength of the wave?(b) What is the tension of the string?(c) What is the transverse speed of the wave at x = 1 m, t = 1 s?(d) What is the maximum transverse speed of the wave?

The power of a sinusoidal wave on a string is given by: P = 0.5 µ2A2.(e) Calculate the increase in amplitude necessary to increase the power of the wave by a factor of 10.

Explanation / Answer

Given

y(t) = a sin(bx ct + 2)

This is in the form of

y(t) = A sin(kx omega*t + 2)

y(t) = 0.01 sin(3x 4t + 2)

a) f = omega/2pi

f = 4/2pi

f = 0.64 Hz

k = 2pi/lambda = 3

lambda = 2pi/3

lambda = 2.09 m

b)

Tension T = v^2 * mue

here mue = 4 kg/m,

v = omega / k = 4/3 = 1.33

T = sqr(1.33) * 4

T = 7.07 N

c)

v(t) = dY/dt = -a*c cos(3x-4t +2)

at t=1s and x = 1m

v = - 0.01*4 cos(3*1 - 4*1 +2)

v = - 0.022 m/s

d)

vmax = a*c = 0.01 * 4

vmax = 0.04 m/s

e)

here A1 / A2 = sqrt(P1/P2)

A2 = sqrt(10) * A1

A2 = 3.16 times of A1

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