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The equation of the vibration of a wire ...

The equation of the vibration of a wire is ` y = 5 "cos"(pi x)/(3) sin 40 pi t ` , where x and y are given in cm and t is given in s. calculate the
velocity of a particle at x = 1 . 5 cm at the instant ` t = (9)/(8)s *`

Text Solution

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` y = 5 "cos"(pir)/(3) sin 40 pi t = (5)/(2) * 2 sin 40 pi t cos (pir)/(3)`
` = (5)/(2) [ sin ( 40 pi t + (pir)/(3))+ sin ( 40 pi t - (pir)/(3))]`
`= (5)/(2) sin 40 pi (t + (x)/(120))+(5)/(2)sin 40 pi (t - (x)/(120))`
` y _(1) + y_(2)`
So, the resultant vibration is produced due to the superposition of two waves ` y _(1) and y_(2)` . comparing these two waves with the general equation, y ` = A sin omega (t - (x)/(V)) ` .
Particl velocity .
` v = (dy)/(dt) = 5 "cos" (pi r)/(3)* 40 pi cos 40 pi t `
` = 200 pi "cos" (pi r) /(3) cos 40 pi r t `
So, at ` x = 1 . 5 cm and t = (9)/(8) s` .
` v = 200 pi "cos" (pi xx 1 . 5 ) /( 3) cos (40 pi xx (9)/(8)) `
` = 200 pi " cos " (pi)/(2) cos 45 pi = 0 ` .
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