A standing-wave pattern is observed in a thin wire with a length of 3.00 m. The wave function is y = 0.00200 sin(πx) cos(100πt) where x and y are in meters and t in seconds.
(a) How many loops does this patten exhibit? (b) What is the fundamental frequency of vibration of the wire?
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The general wave function of a standing wave in a wire is:
y = 2A sin(kx) cos(ωt) comparing this general function with
the given function we get:
k = π =2πλ or λ = 2.00 m and ω = 100π = 2πf or f = 50.0 Hz
(a) The distance between adjacent nodes dNN = λ/2
and the number of loops N on the wire
with length L = 3.00 m is:
N =L/dNN=2L/λ=(2 × 3.00)/2.00
(b) the speed of the
wave in the wire is:
v = fλ = 50.0 × 2.00 = 100 m/s
The wavelength is:
λ = 2L = 6.00 m
This wave travels the wire with the same speed as the wave
that fits 3 loops in the length
of the wire, so fundamental frequency is then:
f =vλ=(100 m/s)/6.00 m
= 16.7 Hz
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