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प्रश्न
For the harmonic travelling wave y = 2 cos 2π (10t – 0.0080x + 3.5) where x and y are in cm and t is second. What is the phase difference between the oscillatory motion at two points separated by a distance of What is the phase difference between the oscillation of a particle located at x = 100 cm, at t = T s and t = 5 s?
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उत्तर
Given, wave functions are y = 2 cos 2π (10t – 0.0080x + 3.5)
= 2 cos(20πt – 0.016πx + 7π)
Now, the standard equation of a travelling wave can be written as y = a cos(ωt – kx + `phi`)
On comparing with the above equation, we get
a = 2 cm
ω = 20π rad/s
k = 0.016π
Path difference = 4 cm
T = `(2π)/ω = (2π)/(20π) = 1/10`s
∴ At x = 100 cm, t = T
`phi`1 = 20πT – 0.016π(100) + 7π
= `20π(1/10) - 16π + 7π`
= 2π – 1.6π + 7π .......(i)
Again, at x = 100 cm, t = 5s
`phi`2 = 20π(5) – 0.016π(100) + 7π
= `100π - (0.016 xx 100)π + 7π`
= 100π – 1.6π + 7π .......(ii)
∴ From equations (i) and (ii), we get
Δ`phi` = Phase difference = `phi_1 - phi_2`
= (100π – 1.6π + 7π) – (2π – 1.6π + 7π)
= 100π – 2π
= 98π rad
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