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151.

An organ pipe `P_(1)` closed at one end vibrating in its first overtone and another pipe `P_(2)` open at both ends vibrating in third overtone are in resonance with a given tuning fork . The ratio of the length of `P_(1)` to that of `P_(2)` isA. `8//3`B. `3//8`C. `1//2`D. `1//3`

Answer» Correct Answer - B
` 3 xx (v)/( 4 l_(c)) = 4 xx (v)/( 2 lo) or (l_(c ))/(lo) = ( 3v)/( 4) xx (2)/( 4 v) = (3)/(8)`
Discussion
152.

A `3 m` long organ pipe both at both ends is driven to third harmonic standing wave . If the amplitude of pressure oscillation is `0.1%` of the mean atmospheric pressure `(P_(0) = 10^(5) N//m^(2))`. Find the amplitude of i. particle oscillation and ii. density oscillation. Speed of sound ` v = 330 m//s , density of air rho_(0) = 1.0 kg//m^(3)`

Answer» Correct Answer - (i) `S_(0) = (1)/(1089 pi) m , (ii) (1)/(1089) kg//m^(3)`
` 3 = 3(lambda)/(2)`
`lambda = 2 m`
`P_(m) = 100 N//m^(2) , V = 330 m//s , rho_(0) = 1 kg//m^(3)`
i. `P_(m) = Bs_(0) k = rho_(0) v^(2) s_(0) ( 2pi)/( lambda)`
`s_(0) = (lambda P_(m))/( rho_(0) v^(2) 2 pi) = ( 2 xx 100)/( 1 xx 330 xx 330 xx 2 pi)`
ii.`B = -( dp)/( dV//V) = (dp)/( d rho//rho)`
`[ m = rho v rArr 0 = (d rho)/(rho) + ( d v)/( v)]`
`rArr d rho = (rho d p)/(B)`
`( d rho)_(max) = ( rho p_(m))/( rho v ^(2)) = (100)/(108900) kg//m^(3) = (1)/( 1089) kg//m^(3)`
Discussion