Question

A transverse sinusoidal wave on a string has a period T = 33.0 ms and travels in the negative x direction with a speed of 30.0 m/s. At t = 0, a particle on the string at x = 0 has a transverse position of 2.00 cm and is traveling downward with a speed of 3.00 m/s. (a) What is the amplitude of the wave? m (b) What is the phase constant? rad (c) What is the maximum transverse speed of the string? m/s (d) Write the wave function for the wave. (Use the form Asin(kx + ωt + ϕ). Round all coefficients to three significant figures.) y(x, t) =

Answer 1

a)

T = time period of the wave = 33 ms = 0.033 sec

angular frequency is given as

w = 2/T = 2 (3.14)/0.033 = 190.3 rad/s

equation for the wave is given as

y(x,t) = A Cos(kx + wt + )

at x = 0

y(t) = A Cos( wt + )

at t = 0

0.02 = A Cos( w(0) + )

0.02 = A Cos                                                  eq-1

velocity is given as

v(t) = dy(t) /dt = (d/dt) (A Cos( wt + )) = - Aw Sin(wt + )

at t = 0

- 3 = - Aw Sin( )

3 = Aw Sin                                                    

3 = A(190.3) Sin      

0.0158 = A Sin                                                     eq-2

dividing eq-2 by eq-2

A Sin /(A Cos ) = 0.0158/0.02

tan = 0.79

= 38.3 deg = 0.67 rad

using eq-1

0.02 = A Cos38.3

A = 0.0255 m

b)

= 38.3 deg = 0.67 rad

c)

maximum speed is given as

V_max = A w

V_max = (0.0255) (190.3) = 4.85 m/s

d)

k = w/v = 190.3/30 = 6.34

equation for the wave is given as

y(x,t) = A Cos(kx + wt + )

equation for the wave is given as

y(x,t) = (0.0255) Cos(6.34x + 190.3t + 0.67)