5) A string oscillates according to the equation
y´ = (0.734 cm) sin[(pi/3.0 cm^-1)x] cos[(22.7 pi s^-1)t].
What are the (a) amplitude and (b) speed of the two waves
(identical except for direction of travel) whose superposition
gives this oscillation? (c) What is the distance between nodes? (d)
What is the transverse speed of a particle of the string at the
position x = 2.27 cm when t = 1.19 s?
Give your answers in centimeter-based units.
Background of question: From the equation and the questions, I
think this is the equation of a standing wave. See
wiki for more info.
a) Since x and t are unrelated variables (or so I think),
amplitude = 0.734cm
b) This one I base my equations on Q4 as kindly answered by
angelfairy.
2π/λ = π/3, 2π/T = 22.7π
λ=6cm, and T = 2/22.7 = 0.088s
thus, f = 1/T = 11.35 s-1
speed = fλ = 68.1 cm/s
c) Nodes are where y is always 0. This happens at the point
where the sin component is 0 since it is dependent on x, the
location of the wave.
so, [ π/3.0 cm^(-1) ] x = 0 or π (2 ajacent nodes. You can try π
and 2π also, just more tedious)
x = 0 or x = 3
so distance = 3-0 = 3cm
d) dy/dt = (0.734 cm) sin[(pi/3.0 cm^-1)x] ( -sin[(22.7 pi
s^-1)t] )
substitute values x = 2.27 cm when t = 1.19 s
should be work in radians I guess
dy/dt = 0.0207 cm/s
I not very sure for c and d
