A standing wave is created in a string which is represented by y=6sin(100πt)cos(5πx). If mass per unit length of string is 100 g/m. Then tension in the thread is
Answers
Answer:y (x , t ) = 2 Sin πx Cos 100π t
x and y are in meters and t is in seconds.
we use the formula in trigonometry 2 Sin A Cos B = Sin (A+B) + Sin (A-B)
y (x, t) = 2 Sin (π x) Cos (100 π t)
= Sin (πx + 100πt) + Sin (πx - 100 πt)
= Sin (πx + 100πt) - Sin (100πt - πx )
= Sin (100 πt + πx) + Sin (100 πt - πx + π)
general formula for a standing wave : y (x, t) = A Sin (ω t - k x + Ф)
these are the component waves which are part of the stationary wave.
component wave 1: y1 (x,t) = sin (π x + 100 π t)
angular frequency = ω = 100 π radians/sec
frequency = f = 50 Hz = ω/2π Time period: 1/f = 0.02 Sec.
Amplitude = A = 1 m
wave number k = - π rad/meter we have formula ω = k v
velocity v = ω / k = - 100 meters/sec
wavelength λ = v / f = 100/50 meters = 2 meters
this component of the wave is traveling in the negative x direction. so its velocity is negative.
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component wave 2: y1 (x,t) = sin ( 100 π t - π x + π )
angular frequency = ω = 100 π radians/sec
frequency = f = 50 Hz = ω/2π Time period: 1/f = 0.02 Sec.
Amplitude = A = 1 m
wave number k = π rad/meter we have formula ω = k v
velocity v = ω / k = 100 meters/sec
wavelength λ = v / f = 100/50 meters = 2 meters
initial phase angle = π radians
this wave is traveling in the positive x direction. so k and v are positive.
But the waves are have a phase difference also.