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A cylindrical tube with a frictionless p...

A cylindrical tube with a frictionless piston ends in a small hole of area of cross–section `2xx10^(-6)m^(2)`, which is very small as compared to the area of cross-section of the tube. The tube is filled with a liquid of density `10^(4)kg//m^(3)`. The tube is held horizontal and the piston is pushed at a constant speed so that liquid comes out of the hole at a constant rate `'Q' m^(3)//s`. No liquid leaks around the piston. The material of the tube can bear a maximum pressure `6xx10^(5)` Pa. The maximum possible value of Q so that the tube does not break is __________ `xx10^(-5)m^(3)//s`. [Atmospheric pressure `= 10^(5)` Pa]

Text Solution

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The correct Answer is:
2
Let atmospheric pressure be `P_(o)`. Suppose the pressure inside the tube is `6P_(o)` (the maximum allowed) and the liquid comes out of the hole at speed v.
Since the hole is very small, the velocity of the liquid inside the tube will be negligible
Applying Bernoulli’s theorem,
`6P_(o)=P_(o)+(1)/(2)rhov^(2)impliesv=sqrt((10P_(o))/(rho))=sqrt((10xx10^(5))/(10^(4)))=10m//s`
Therefore, `Q_(max)=vxx` (area of cross-section of the hole)
`impliesQ_(max)=(10)(2xx10^(-6))=2xx10^(-5)m^(3)//s`
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