Question

A transverse sinusoidal wave on a string is moving in the −x-direction. Its speed is 30.0 m/s, and its period is 20.0 ms. At t = 0, a colored mark on the string at x = 0 has a vertical position of 2.00 cm and is moving down with a speed of 1.30 m/s. (a) What is the amplitude of the wave (in m)? 0.0204 Correct: Your answer is correct. m (b) What is the phase constant (in rad)? Incorrect: Your answer is incorrect. Use the expressions you wrote for y and vtransverse from part (a), and divide them to find an equation for tan(ϕ). Take the inverse to find ϕ. Note that the angle you are looking for is in the second quadrant, so you will need to modify your calculator answer as needed. rad (c) What is the maximum transverse speed of the string (in m/s)? m/s (d) Write the wave function for the wave. (Use the form A sin(kx + ωt + ϕ). Assume that y and x are in m and t is in s. Do not include units in your answer.) y(x, t) = m

Answer 1

A wave traveling in -x direction is given by

y(t) = A Sin[(t+x/) + ]

A - amplitude

= 2/T = 0.314 e+3

= 30 m/s

at x=0 , t=0 y = 0.02

putting these values  

0.02 = A Sin

dy/dt = A Cos[(t+x/) + ]  

at x=0 , t =0

dy/dt|_t=0,x=0 = A Cos[ ] = -1.3

Tan = 4.769

Phase const. = 78.16 deg = 1.364 rads

angle in 2nd quadrant = 2.935 rads

A = 0.0204

max. transverse speed dy/dt|_max = A = 0.0204 * 0.314e+3 = 6.406 m/s

k = / = 0.314e+3 /30 = 10.47

wave eq.

y(t) = ASin( kx +t + ) = 0.0204 Sin(10.47 x + 314 t + 2.935)